Transfer operator for a hyperbolic equation. Numerical methods for solving partial differential equations of hyperbolic type (for example, the transport equation)

Consider the Cauchy problem for an equation of the form

in which the transfer rate v could be a function x. For equation (6.1), one can propose a variety of difference schemes that differ in the order of approximation, the way of representing the derivatives, etc. Let us first dwell on explicit difference schemes, in which each equation of the system contains only one unknown quantity) ", which allows us to successively calculate the values \u200b\u200bof the solution on a new time layer.

It is known that the most important property that should be possessed by explicit difference schemes is stability, the ability of a scheme not to accumulate computational disturbances. The stability of the scheme is a necessary requirement to ensure the convergence of the difference solution to the exact one. For a hyperbolic equation, stability analysis is usually performed with respect to the initial data on the basis of the spectrum of eigenvalues \u200b\u200bof the operator of transition to a new time layer, on the basis of which difference schemes acceptable for calculations are selected. Thus, the symmetric difference scheme

has a very stringent stability condition (m 2 vh) and ns is used for practical algorithms. Difference schemes


are conditionally stable. To ensure their stability, it is necessary, firstly, the fulfillment of the Courant Friedrichs-Levy (CFL) condition:

and secondly, the use of differences towards the flow, i.e. application of scheme (6.3) for V \u003e 0 and (6.4) for v 0.

Explicit scheme with upstream differences. If we selectively apply the two previous schemes, namely, for v\u003e\u003e 0 scheme (6.3), and for v

will be indifferent to the direction of speed and stable under the condition v / h ^ 1. It is easy to see that one-sided differences in this scheme are taken towards the flow (they say that the scheme has the property mpanenopmuenoemu). Schemes) "of this type is called countercurrent or circuit with differences upstream.

In the case of an equation with a constant value of the transfer rate, there are no problems with the design of the upwind difference scheme. The difference corresponding to the sign of the transfer speed is selected and used at all nodes of the computational domain. Condition (6.5) imposes a restriction on the ratio of the steps of the computational grid. Usually, for a given step in space, the admissible time step t h / v is determined from relation (6.5).

But if the transfer rate is a function of the coordinate (or time), then the choice of the type of difference approximation must be based on the analysis of the transfer rate sign, for example, using a conditional operator. Except got, at variable transfer rate v \u003d v (x) the stability condition must be checked for all grid nodes and from this set of values \u200b\u200bof the time step choose the minimum one: t min; h / vj.

In the work of Courant et al. (1952), an interesting method for constructing an upstream scheme was proposed, in which the conditional operator was not used. It is important to note that this is not just a formal technique, but an approach containing deep ideas on the basis of which one can compare and find a correspondence between upwind (asymmetric) and symmetric difference schemes. The idea of \u200b\u200bsplitting the operators of difference schemes is close to this.

Let us represent the transfer rate as the sum of its positive and negative components:

This will allow you to represent the carry operator as the sum of two operators:

Now each of the operators has a constant sign coefficient, which makes it possible to apply upwind difference approximation to it. Note that the upstream difference scheme for approximating convective terms is widely used in various problems of computational fluid dynamics. The following notation of the computational algorithm according to the scheme (6.6) is often used:

If we now carry out elementary transformations on the right-hand side of (6.7) and select the symmetric difference derivative, then this scheme will be represented in the form

It can be concluded that the counterflow difference scheme (6.7) is equivalent to symmetric (6.2), into which a dissipative additive is introduced, which ensures the conditional stability of the scheme.

Lax's scheme. This scheme was introduced into the practice of computing at the dawn of the development of computational gas dynamics. II although references to a scheme of this type were found in the works of various authors, public opinion associates it with the name of the American mathematician Lax (P.D.), who published in the 1950s a series of works on various aspects of the theory of difference schemes. As applied to the transport equation (6.1), this scheme has the form

A feature of the scheme is that to ensure its stability in the approximation of the time derivative, the value of the grid function at the node (r, p) is replaced with a half-sum of values \u200b\u200bin adjacent nodes of the same time layer. This operation ensures, with the central approximation of the spatial derivative, the conditional stability of the difference scheme (under the Courant - Friedrichs - Levy condition v / h ^ 1).

Although here the derivative with respect to x is presented with a second order of approximation, the scheme has significant dissipation due to the specific representation of the time derivative. This is clearly seen from the first differential approximation:

The coefficient on the right in front of the second derivative can be interpreted as the coefficient of schematic viscosity. After simple transformations, this value can be represented as

where through and the Courant number is indicated. Many properties of this circuit can be determined from the differential approximation:

  • - the scheme becomes nondissipative when the Courant number is equal to one;
  • - the circuit is not sensitive to the direction of flow;

when the Courant number is less than one, the circuit viscosity has a stabilizing effect (positive diffusion coefficient); when the Courant number is greater than one, the circuit viscosity becomes negative, which leads to an aggravation of the diffusion process and, ultimately, to the loss of the computational stability of the circuit;

As the time step decreases, the dissipative properties of the circuit increase.

Among the listed features, there are those that significantly reduce the advantages of the circuit. However, the simplicity of the algorithm is often the basis for its use at the initial (debugging) steps of building computational programs. In addition, the Lax scheme, as we will see below, is an integral part of efficient multistep algorithms, in which a preliminary step (forecast step) is performed with its help.

Second order schemes. The difference schemes discussed earlier were of the first order (in spatial or temporal variable). When constructing second-order schemes, it is necessary to provide an increased order of approximation both in spatial and temporal variation. Let's consider several schemes of this type.

Leapfrog scheme. A second-order scheme both in the spatial variable and in time of the simplest type can be represented as

This scheme is called a stepping scheme, but it is better known as "leapfrog" (leap-frog scheme). The schematic is three-layer and builds the solution from the two previous time layers. Therefore, when using it, problems arise with the start of calculations, which must be carried out by some other method.

Lax-Wendroff scheme. One of the most famous schemes of this type is the central scheme, named after its authors, the Lax-Wendroff scheme. It has occupied a certain niche in the theory of difference schemes for hyperbolic equations, many very productive ideas are associated with it, but its main advantage is that it can be easily generalized and transferred to the case of more complex problems - problems of compressible gas flow described by systems of quasilinear equations, where it has been one of the main computational tools for a long time.

It is useful to study the features of this scheme by an example of its application to a transport equation of the form (6.1). To construct a second-order circuit, we write out the Taylor formula:

which we will consider together with the original equation (6.1) This equation will be used in order to replace the time derivatives in the expansion with spatial ones. This is possible, since the first time derivative is expressed directly from (6.1): du / dt \u003d -vdu / dx. The second derivative is also easily found from the following chain of relations:

Note that this representation is accurate only at a constant transfer rate: v \u003d const. Otherwise, it is approximate, however, if the transfer rate v (x) a sufficiently smooth function, it can be used to transform difference relations that are local in nature.

Substituting the expressions for the derivatives obtained using the original differential equation into the above Taylor formula, we obtain the relation

and replacing the derivatives with respect to space by second-order finite-difference relations, we obtain (after some simple transformations) the difference scheme

called the Lax Wendroff scheme. This scheme was introduced into computational practice along with a number of others in a series of papers published by Lax and Wsndroff in 1960-1964.

A two-step version of the Lax-Wendroff scheme. Later Richtmeier proposed an original two-step version of the scheme, which, due to the convenience in implementation, was for a long time one of the main computational algorithms for gas dynamics. Let's give this option.

In the first half step, we calculate the intermediate value of the solution using a simple first-order Lax scheme. This intermediate value is assigned a superscript n + 1/2 and we will keep in mind that a half time step is also used. Applying this scheme, we get the solution values \u200b\u200bat the intermediate time layer: t \u003d t n + l / 2. Note that, due to the use of the Lax scheme, in which there is no central node on the lower layer, the solution is reproduced on the intermediate layer also in a system of half-integer points.

Here is a record of the difference relations for two adjacent intervals:


The second half-step consists in calculating the solution on a new time level p + 1 based on a scheme with central differences in both space and time - the "cross" scheme. To calculate the spatial derivatives, the values \u200b\u200bof the solution on the intermediate layer in the system of half-integer points are used, the solution itself is reconstructed in the same system of points in which it was determined at the beginning of the time step:

Relations (6.12) and (6.13) together define the two-step Lax - Weidroff scheme. At its first stage, the fulfillment of stability conditions is ensured. This stage is sometimes called predictor. The second stage ensures that the required accuracy is achieved, and it is called proofreader. Predictor-corrector methods are often used in computational mathematics, and the corrector step may include an iteration block.

It can be easily shown that, excluding intermediate values \u200b\u200bfrom (6.13), using relations (6.12) we arrive at the main - one-step - variant of the scheme. In the sense of the order of approximation and stability, both options are equivalent, but the two-step one is more convenient when carrying out calculations, therefore, the name of this difference scheme is usually associated with it. The two-step version is especially convenient for constructing difference schemes for more complex problems, in particular, for systems of quasilinear equations of nonstationary gas dynamics.

Monotonicity of the solution in second-order schemes. The last term on the right-hand side of (6.11) has a form different from the form of the dissipative terms of the first-order schemes (6.8) and (6.10). In this case, it provides suppression of the error associated with the first order of approximation of the time derivative. Thus, this scheme is a second-order scheme of both time and space variable. Its first differential approximation will no longer contain a dissipative term, but it will contain a dispersion component with a third derivative, which is the cause of phase errors in the circuit. It can be expected that this scheme will slightly smear the solution, but nonphysical oscillations caused by dispersion can appear in the region of its sharp change.

A difference scheme that transforms a solution in the form of a monotonic function of the longitudinal coordinate into a monotonic solution is called monotonic difference scheme. According to this definition, the Lax-Weidroff scheme is non-monotonic.

S.K. Godunov established the monotonicity theorem, which occupies one of the central places in the theory of difference schemes. According to this theorem, for a linear equation of the form (6.1), there are no monotone schemes with an order higher than the first.

The loss of monotonicity of the difference scheme is characteristic, to one degree or another, for all schemes of increased approximation order. To overcome the nonmonotonicity of the numerical solution of high-order schemes, the so-called hybrid difference schemes. They belong to the class of nonlinear ones, in which, based on the analysis of the solution behavior, switching to monotonic first-order schemes in zones where phase errors are especially pronounced, and returning to high-order schemes in the regions of smooth solution change are performed.

McCormack's scheme. It is also a second-order two-step scheme that is indifferent to flow direction. It is more convenient to demonstrate it in the conservative form of the transport equation:

The scheme consists of two sequential steps:


At the first stage (6.15), the preliminary value of the solution is found u at grid points based on a one-way difference scheme. Based on the solution found in this way, the preliminary values \u200b\u200bof the fluxes / r are calculated. Further, on the basis of one-sided schemes with the opposite direction (6.16), the solution is determined at the next time layer.

This algorithm allows various modifications, it adapts well to solving both quasilinear systems and multidimensional hyperbolic problems. In the 1970s, this scheme was one of the main difference schemes of foreign (mainly American) computers, but now it has been superseded by more modern ones based on the ideas of hybridization.

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2 MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION NOVOSIBIRSK STATE UNIVERSITY Faculty of Mechanics and Mathematics Department of Mathematical Modeling GS Khakimzyanov, SG Cherny CALCULATION METHODS Part 4. Numerical methods for solving problems for hyperbolic equations Textbook Novosibirsk 014

3 BBK V.193 UDC X 16 Reviewer Cand. physical-mat. Sciences AS Lebedev The publication was prepared as part of the implementation of the Development Program of the state educational institution of higher professional education "Novosibirsk State University" for years. X 16 Khakimzyanov, GS Methods of computation: In 4 hours: textbook. allowance / G. S. Khakimzyanov, S. G. Cherny; Novosib. state un-t. Novosibirsk: RITs NSU, 014. Part 4: Numerical methods for solving problems for equations of hyperbolic type. 07 p. ISBN The textbook corresponds to the program of the course of lectures "Methods of Computation", which is read at the Faculty of Mechanics and Mathematics of NSU. In its fourth part, the foundations of numerical methods for solving initial-boundary value problems for equations of hyperbolic type are presented, problems for seminars are formulated, samples of tests and tasks for practical exercises on a computer are given. The manual is intended for students and teachers of mathematical specialties of higher educational institutions. ISBN LBC V.193 UDC c Novosibirsk State University, 014 c G.S. Khakimzyanov, S. G. Cherny, 014

4 CONTENTS Preface Schemes for the linear transport equation Monotonicity property of difference schemes Construction of monotonic schemes based on the differential approximation method Schemes for a nonlinear transport equation Schemes on an adaptive grid for the transfer equation Difference schemes for the equation of string vibrations Difference schemes for a hyperbolic system of equations with constant coefficients Difference schemes for systems of nonlinear equations of shallow water Difference schemes for problems of gas dynamics Control work on the topic "Investigation of difference schemes for the transport equation" Tasks for laboratory work Answers, instructions, solutions Bibliography

5 Preface In the fourth part of the manual, the foundations of numerical methods for solving initial-boundary value problems for equations of hyperbolic type are stated, problems on this topic for seminars are formulated, tasks for practical exercises on a computer and an example of control work are given. Theoretical questions are presented rather briefly. For a deeper study of the issues under consideration, we recommend referring to the textbook of S.K. Godunov and V.S. Ryabenky, as well as to the books of G.I. Marchuk, A.A. Samarsky, A.A. Samarsky and A.V. Gulin , A. A. Samarsky and E. S. Nikolaev, B. L. Rozhdestvensky and N. N. Yanenko and textbooks published at NSU. The lectures deal with theoretical issues related to the study of only finite-difference schemes. As examples, schemes for a linear transport equation, a nonlinear scalar first-order equation, a second-order equation describing string vibrations, a linear system of first-order equations, a system of nonlinear shallow water equations, and equations of gas dynamics are considered. Each paragraph is accompanied by tasks that need to be solved at the seminar. Many tasks are provided with instructions and detailed solutions. Additional materials for seminars can be found in problem books. The manual provides examples of tasks for practical exercises in computer classes, gives recommendations on how to complete tasks, discusses issues related to the development of programs and the presentation of results. Additional tasks can be taken from teaching aids. The fourth part of the manual has independent continuous numbering of paragraphs and figures and an independent bibliographic list. Inside the paragraphs for formulas and statements (lemmas and theorems), two-index numbering is used, for example 4 .. References to formulas, lemmas, theorems from the previous three parts of the manual are given by adding the number 1 or 3 to their number in front of their number. For example, instead of “according to the formula ( 4.) from the manual "we write" according to formula (1.4.) ", Instead of" according to Theorem 8.3 from the manual "" according to Theorem 8.3 ". The authors are deeply grateful to the referee Alexander Stepanovich Lebedev for valuable advice and critical comments that contributed to the improvement of this textbook. 4

6 1. Schemes for the linear transport equation 1.1. Some information from the theory of hyperbolic systems. Consider the Cauchy problem for a linear system of first-order differential equations u t + A u \u003d f (x, t),< x <, 0 < t T, x u(x, 0) = u 0 (x), < x <. (1.1) Здесь u = (u 1,..., u m) T m-мерная вектор-функция переменных x, t, A вещественная m m матрица с элементами a i (x, t). Определение. Систему уравнений (1.1) будем называть гиперболической в некоторой области переменных (x, t), если в каждой точке этой области собственные значения λ 1, λ,..., λ m матрицы A вещественны и различны. Определение. Интегральная кривая x = x k (t) обыкновенного дифференциального уравнения dx dt = λ k(x, t) (1.) называется k-ой характеристикой системы уравнений (1.1). Предполагается, что элементы матрицы A обладают гладкостью, достаточной для того, чтобы через каждую точку плоскости (x, t) проходила единственная характеристика, отвечающая собственному значению λ k. Характеристики, проведенные через точку (x, t) (t > 0) in the direction of decreasing time t, will cross the Ox axis at m different points. Let us arrange the eigenvalues \u200b\u200bof the hyperbolic system (1.1) (λ 1 (x, t)< λ (x, t) <... < λ m (x, t)) и через обозначим отрезок оси Ox, ограниченный точками пересечения этой оси с m-ой и первой характеристиками. Определение. Областью зависимости точки (x, t) для системы уравнений (1.1) называется множество точек верхней полуплоскости, ограниченное крайними характеристиками x = x m (t), x = x 1 (t) и отрезком . Область зависимости точки (x, t) изображена на рис. 1, а. Решение u системы (1.1) в точке (x, t) будет зависеть только от значений u 0 (x) на 5

7 segment. Therefore, if the initial data outside the segment are changed to others, then the solution at the point (x, t) will not change. Definition. The domain of influence of a point (x 0, 0) is the set of points (x, t) of the upper half-plane, bounded by the extreme characteristics of system (1.1) outgoing from (x 0, 0), i.e., the characteristics corresponding to the eigenvalues \u200b\u200bλ 1 and λ m. The area of \u200b\u200binfluence of the point (x 0, 0) is shown in Fig. 1, b. If the initial data are changed only at the point (x 0, 0), then the solution of the hyperbolic system will change only at the points (x, t) belonging to the influence area of \u200b\u200bthe point (x 0, 0). Suppose now that instead of the Cauchy problem (1.1) we need to solve an initial-boundary value problem on an interval. Then, in addition to the initial conditions, boundary conditions must be specified. The number of boundary conditions at each of the boundaries is determined by the number of characteristics entering the area. For example, if m 0 characteristics enter the region through the left boundary x \u003d 0, ie, m 0 eigenvalues \u200b\u200bλ k are positive at x \u003d 0, then m 0 boundary conditions must be specified on this boundary. If on the boundary x \u003d l the number of negative eigenvalues \u200b\u200bis equal to m l and, therefore, exactly m l characteristics enter the region through the right boundary, then at this boundary it is necessary to set m l boundary conditions. Since the eigenvalues \u200b\u200bdepend on time, the number of boundary conditions at each of the boundaries can change with time. t dx dt \u003d m λ m (x, t) dx dt \u003d λ 1 t dx dt \u003d λ 1 dx dt \u003d m λ x l a x r x (x 0,0) b x Fig. 1. Characteristics of the system of equations (1.1) limiting the regions of dependence of the point (x, t) (a) and the influence of the point (x 0, 0) (b) 6

8 Consider now the homogeneous hyperbolic system of equations (1.1) with constant coefficients. For a constant matrix A, its eigenvectors and eigenvalues \u200b\u200bare constant, i.e., they do not depend on x and t. Let l k be the kth left eigenvector of the matrix A corresponding to its eigenvalue λ k: l k A \u003d λ k l k (k \u003d 1, ..., m). We multiply system (1.1) on the left by the vector lk: or where lkut + l ka ux \u003d 0. This equation can be written in the following form: lkut + λ klkuxskt + λ skkx \u003d 0, \u003d 0, (1.3) sk \u003d lku, k \u003d 1, ... m. (1.4) The solution sk (x, t) of equation (1.3) is transferred along the characteristic without change and therefore is calculated for t\u003e 0 from the initial value sk at the point of intersection of the kth characteristic with the axis Ox: sk (x, t) \u003d sk ( x λ kt, 0). (1.5) The functions s k are called Riemann invariants. 1 .. Linear shallow water model. The simplest mathematical model, within which it is possible to describe the motion of a fluid with surface waves, is a linear model of shallow water: η t + u 0 \u003d 0, (1.6) xut + g η \u003d 0, (1.7) x η (x, 0) \u003d η 0 (x), u (x, 0) \u003d u 0 (x), (1.8) where η (x, t) is the rise of the liquid surface above the unperturbed level (see Fig.), u (x, t) is the liquid velocity , η 0 (x) and u 0 (x) the elevation and velocity at the initial time moment t \u003d 0, 0 \u003d const is the depth of the basin, g \u003d const is the acceleration of gravity. 7

9 The system of equations (1.6), (1.7) can be written in the form of a homogeneous system (1.1) with a matrix A and a solution vector u: A \u003d (0 0 g 0) (η, u \u003d u). (1.9) The matrix A has two different real eigenvalues \u200b\u200bλ 1 \u003d c 0, λ \u003d c 0 \u003d g 0, (1.10) therefore the system of equations (1.6), (1.7) is of hyperbolic type. Equations of characteristics (1.) take the following form: dx dt \u003d c 0, dx dt \u003d c 0, (1.11) therefore the characteristics are straight lines. The characteristics passing through the point (x, t), t\u003e 0, intersect the Ox axis at the points x l and x r, where x l \u003d x c 0 t, x r \u003d x + c 0 t. (1.1) The left eigenvectors of the matrix A corresponding to the eigenvalues \u200b\u200b(1.10) are given by the formulas l 1 \u003d (c 0, 0), l \u003d (c 0, 0). (1.13) y 0 η y \u003d (x, t) lxy \u003d - 0 Fig. Notation in the problem of propagation and transformation of waves in a basin with vertical walls According to (1.4), the relationship between the Riemann invariants r \u003d s 1, s \u003d s and initial dependent variables is given by the formulas r \u003d c 0 η 0 u, s \u003d c 0 η + 0 u, (1.14) 8

10 whence η \u003d r + sc 0, u \u003d sr 0. (1.15) From formula (1.5), taking into account equalities (1.14), we obtain formulas for solving the Cauchy problem in invariants r (x, t) \u003d r (x λ 1 t, 0) \u003d r (x + c 0 t, 0) \u003d c 0 η 0 (xr) 0 u 0 (xr), (1.16) s (x, t) \u003d s (x λ t, 0) \u003d s (xc 0 t, 0) \u003d c 0 η 0 (xl) + 0 u 0 (xl). (1.17) Finally, using relations (1.15), we obtain an exact solution to the Cauchy problem (1.6), (1.7), (1.8) η (x, t) \u003d η 0 (xl) + η 0 (xr) + 0 u0 ( xl) u 0 (xr), c 0 u (x, t) \u003d u 0 (xl) + u 0 (xr) + c 0 η0 (xl) η 0 (xr). 0 (1.18) When solving the initial-boundary value problem under consideration, it is necessary to set one condition at each endpoint of the segment. For example, we will assume that the walls of the pool are impermeable to the liquid, which means that the velocity of the liquid on these walls is equal to zero: u (0, t) \u003d u (l, t) \u003d 0. (1.19) Let us give the final form of the mathematical formulation of the problem on the motion of a fluid with surface waves in a bounded basin: find a solution η (x, t), u (x, t), continuous in a closed domain D \u003d, to the following initial-boundary value problem η t + u 0 x \u003d 0, ut + g η \u003d 0, 0< x < l, 0 < t T, x u(0, t) = u(l, t) = 0, 0 t T, η(x, 0) = η 0 (x), u(x, 0) = u 0 (x), 0 x l. (1.0) 1.3. Линейное уравнение переноса. Итак, если матрица A однородной гиперболической системы уравнений (1.1) постоянна, то такую систему можно свести к системе уравнений в инвариантах Римана, 9

11 in this case, the equations for the Riemann invariants are independent of each other and each of them has the form u t + au x \u003d 0, a \u003d const. (1.1) This equation is the simplest hyperbolic equation and is called the linear transport equation. This equation can be used to study the properties of difference schemes used to solve hyperbolic systems of equations. Consider for the linear transport equation (1.1) the Cauchy problem u t + au x \u003d 0,< x <, 0 < t T, u(x, 0) = u 0 (x), < x <. (1.) Характеристика x = x(t) уравнения (1.1) определяется уравнением dx dt = a, (1.3) т. е. является прямой с наклоном a к оси Ot. Следовательно, точное решение задачи Коши определяется по формуле u(x, t) = u 0 (x at). (1.4) График точного решения в момент времени t получается переносом графика начальной функции на величину at (в положительном направлении оси Ox, если a > 0 and vice versa). For the transport equation with a constant coefficient a, it is easy to write down the exact solution for the initial-boundary value problem. Let, for example, a \u003d const\u003e 0. Then the following initial boundary value problem is correct u t + au x \u003d 0, 0< x l, 0 < t T, u(0, t) = µ 0 (t), 0 t T, u(x, 0) = u 0 (x), 0 x l, u 0 (0) = µ 0 (0). (1.5) Легко проверить, что если u 0 (x) и µ 0 (t) дифференцируемые функции, то решение задачи (1.5) определяется формулой u(x, t) = { u0 (x at) при t x/a, µ 0 (t x/a) при t x/a. (1.6) 1.4. Явная противопоточная схема. Перейдем теперь к изучению конечно-разностных схем решения линейного уравнения переноса. 10

12 We start with an explicit scheme with upstream differences (upstream scheme) for the initial boundary value problem u t + au x \u003d f (x, t), 0< x l, 0 < t T, a = const > 0, u (0, t) \u003d µ 0 (t), 0 t T, u (x, 0) \u003d u 0 (x), 0 x l, u 0 (0) \u003d µ 0 (0). (1.7) Throughout what follows, we will consider only uniform grids covering the closed region D \u003d. We construct the following difference scheme un + a un un 1 \u003d fn, \u003d 1, ..., N, un 0 \u003d μ n 0, n \u003d 0, ..., M, u 0 \u003d u 0 (x), \u003d 0 , ..., N, (1.8) approximating problem (1.7) with order O (+). As before, this scheme can be written in the operator form L u \u003d f. The name upwind scheme is associated with the fact that if we consider the transport equation as a model equation for the system of equations describing the flow of a liquid or gas, and we identify the coefficient a with the liquid velocity, then at a positive velocity, i.e., for a\u003e 0, in the scheme the left-hand difference derivatives are taken using the node x 1 located upstream (upstream). We introduce uniform norms in the space of grid functions U and the space of the right-hand sides F: where f F (\u003d max u U max nun C \u003d max 0 N un, \u003d max n un C, (1.9)) µn 0, \u200b\u200b(u 0) C, max fnn C, (1.30) fn C \u003d max 1 N fn uniform norms on the layer t \u003d t n. Using the maximum principle, one can prove the following statement. Theorem 1.1. Satisfaction of condition a 1 (1.31) 11

13 is sufficient for the stability of the upwind scheme (1.8) in a uniform norm. PROOF. Let x be a grid node with number 1 N. Let us rewrite the difference equation of the circuit at this node \u003d (1 r) u n + ru n 1 + f n, where r \u003d a /. The hypothesis of the theorem implies that 1 r 0; therefore, the following estimate will be valid (1 r) un + run 1 + fn (1 r) un C + run C + fn C un C + max mfm C. At the boundary node, we have the following estimate 0 \u003d µ n + 1 0 max m µm 0. Therefore, the maximum of the left-hand sides of these inequalities cannot exceed the maximum of the two numbers on the right-hand sides of these inequalities: (C max max m) µm 0, un C + max fmm C, and this is the maximum principle. We found that under condition (1.31) scheme (1.8) satisfies the maximum principle. Therefore (see Theorem 3.1.1) it will be stable in the uniform norm with respect to the initial data, boundary conditions, and the right-hand side. The same condition (1.31) is also a necessary condition for the stability of scheme (1.8), which follows from the spectral criterion for Neumann stability. Let's prove it. We take the harmonic u n \u003d λ n e iφ (1.3) and substitute it into the homogeneous difference equation. As a result, for the transition factor we obtain the equation Therefore, λ \u003d 1 r (1 e iφ) \u003d 1 r (1 cos φ) ir sin φ. λ \u003d 1 r (1 cos φ) + r (1 cos φ) + r sin φ \u003d 1

14 \u003d 1 r (1 cos φ) [r (1 cos φ) r (1 + cos φ)] \u003d 1 r (1 cos φ) (1 r). Let the steps in scheme (1.8) and are related by the law of passage to the limit r \u003d a \u003d const. (1.33) Then the eigenvalues \u200b\u200bλ (φ) do not depend on, therefore the necessary condition for Neumann stability is reduced to the requirement or λ (φ) 1, φ R. (1.34) r (1 cos φ) (1 r) 0, φ R. (1.35) Obviously, this inequality is equivalent for a\u003e 0 to condition (1.31). Thus, condition (1.31) for a\u003e 0 is a necessary and sufficient condition for the stability of the upwind scheme in the uniform norm. Note that for a< 0 схема (1.8) абсолютно неустойчива, поскольку в этом случае нарушается неравенство (1.34) (см. задачу 1.1). Какую же схему следует использовать при a < 0, когда поток распространяется справа налево? Отметим, что в этом случае корректной будет такая начально-краевая задача u t + au x = f(x, t), 0 x < l, 0 < t T, a = const < 0, u(l, t) = µ l (t), 0 t T, u(x, 0) = u 0 (x), 0 x l, u 0 (l) = µ l (l). (1.36) Для этой задачи возьмем следующую противопоточную схему u n + a un +1 un = f n, = 0,..., N 1, u n N = µn l, n = 0,..., M, u 0 = u 0(x), = 0,..., N, (1.37) которая аппроксимирует дифференциальную задачу (1.36) с порядком O(+). Используя принцип максимума и спектральный признак Неймана, можно показать, что схема (1.37) при a < 0 будет устойчива при выполнении условия a 1. С другой стороны, при a > 0 scheme (1.37) will be absolutely unstable (see problem 1.). 13

15 Thus, we have constructed two conditionally stable explicit schemes with upstream differences for the transport equation with a constant coefficient aunun + a un un 1 + a un +1 un They are stable under the inequality \u003d fn if a\u003e 0, \u003d fn, if a< 0. (1.38) a 1. (1.39) Во внутренних узлах сетки противопоточную схему (1.38) можно записать в виде одного уравнения u n + a + a u n un 1 + a a u n +1 un = f n. (1.40) Аналогично выглядит явная противопоточная схема и в случае знакопеременного коэффициента a(x, t). Например, если на границах отрезка выполнены условия a(0, t) > 0, a (l, t)< 0, 0 t T, то получим такую противопоточную схему где u n + a + un un 1 + a un +1 un = f n, = 1,..., N 1, u n 0 = µ n 0, u n N = µ n l, n = 0,..., M, (1.41) u 0 = u 0 (x), = 0,..., N, a + = an + a n, a = an a n, (1.4) которая аппроксимирует с порядком O(+) начально-краевую задачу u t + a(x, t)u x = f(x, t), 0 < x < l, 0 < t T, u(0, t) = µ 0 (t), u(l, t) = µ l (t), 0 t T, u(x, 0) = u 0 (x), 0 x l. (1.43) 14

16 Using the maximum principle, one can prove (see Problem 1.10) that for the stability of the upwind scheme (1.41) with a variable coefficient a (x, t), it is sufficient to satisfy the condition max a (x, t) 1. (1.44) x, t 1.5 ... Lax's scheme. Further, for simplicity of presentation, we will consider the initial-boundary value problem (1.7) with the homogeneous transport equation ut + au x \u003d 0. (1.45) In the Lax scheme, the difference equation approximating the transport equation (1.45) is written as 0.5 (un +1 + ) un 1 + a un +1 un 1 \u003d 0, \u003d 1, ..., N 1. (1.46) For the local approximation error, we have the expression ψ n, \u003d u tt u xx + ..., therefore, for \u003d O ( ) the Lax scheme will not approximate the transport equation, and under the law of the passage to the limit r \u003d a \u003d const (1.47) will approximate with the order O (+). Thus, the approximation takes place only for a certain connection between the steps and, i.e., the Lax scheme belongs to the class of conditionally approximating schemes. For the transition factor, we obtain the formula λ (φ) \u003d cos φ ir sin φ. Therefore, under the law of passage to the limit (1.47), the necessary condition for the stability of the Lax scheme is the fulfillment of the inequality r 1, i.e., a 1. (1.48) 15

17 1.6. Lax Wendroff's scheme. The difference equations of this scheme look like this u + 1/0, 5 (un +1 +) un + a un +1 un \u003d 0, / un + au + 1 / (1.49) u 1 / \u003d 0. The Lax - Wendroff scheme belongs to family of two-step circuits. In this scheme, at first, at half-integer nodes x + 1 / \u003d x + /, according to the Lax scheme, auxiliary quantities u + 1 / are calculated, which refer to the time instant t n + /. Then, at the second step, the values \u200b\u200bof the required grid function are calculated at the (n + 1) th layer in time. To study the approximation and stability of two-step schemes, the auxiliary quantities u are previously excluded from the scheme. As a result of elimination, we obtain the one-step Lax - Wendroff scheme u n + a un +1 un 1 \u003d a un +1 un + un 1, (1.50) which, as it is easy to verify, approximates the transport equation (1.45) with the second order in and. For the transition factor, we have the following expression λ \u003d 1 ir sin φ r sin φ. Therefore, the necessary condition for the stability of λ 1 will be equivalent to the fulfillment of the inequality (1 r sin φ) + r sin φ 1, or 1 4r sin φ + 4r4 sin 4 φ + 4r sin φ (1 sin φ) 1. The last inequality is equivalent to condition r 1. Thus, the necessary condition for the stability of the Lax-Wendroff scheme coincides with the necessary condition (1.48) for the stability of the Lax-Wendroff scheme Dissipation and dispersion. Along with the transport equation u t + au x \u003d 0, a \u003d const (1.51) 16

18 consider two more equations u t + au x \u003d µu xx, µ \u003d const\u003e 0, (1.5) u t + au x + νu xxx \u003d 0, ν \u003d const. (1.53) Let the initial function in the Cauchy problem for these equations be represented as a Fourier series u (x, 0) \u003d m b m e imx. (1.54) We will seek a solution to each of these equations by the separation of variables u (x, t) \u003d bm λ te imx \u003d bmum (x, t), (1.55) mm where um (x, t) is a harmonic with the wave number mum (x , t) \u003d λ te imx, (1.56) λ is to be determined. The real and imaginary parts of the harmonic are m-waves, the length l of which is related to the wave number by the formula l \u003d π m. (1.57) Since equations (1.51) (1.53) are linear, the behavior of each of the harmonics can be considered independently. Substituting the harmonic with wavenumber m into the transport equation (1.51), we obtain either ln (λ) + aim \u003d 0 λ \u003d e aim. Therefore, if the harmonic (1.56) is a solution to the transport equation, then it has the form Denoting ξ \u003d x at, we obtain u m (x, t) \u003d e im (x at). (1.58) u m (x, t) \u003d e imξ \u003d u m (ξ, 0). (1.59) 17

19 Thus, at any time t\u003e 0, the harmonic u m is obtained by shifting the initial harmonic by at. Consequently, the transfer equation describes the motion of m-waves, which, regardless of their length, propagate at a constant speed v m \u003d a without distorting their shape. It is easy to check that the harmonic (1.56) will be a solution to the second equation (1.5) if ln (λ) + aim \u003d µm or λ \u003d e aim e µm, i.e., the harmonic in this case has the form um (x, t) \u003d e µmt e im (x at). Consequently, for all harmonics, attenuation of the wave amplitude (wave dissipation) occurs. Since m \u003d π / l, short waves decay faster than long ones. The speed v m of wave propagation does not depend on the wavelength and is still equal to a. The term µu xx with the second derivative of the solution is responsible for wave dissipation. Finally, substituting the harmonic into equation (1.53) yields ln (λ) + aim + ν (im) 3 \u003d 0, or whence we obtain λ \u003d e im (a νm), um (x, t) \u003d e im (x ( a νm) t). Thus, the third equation describes the motion of the wave without changing its amplitude (without dissipation). But the speed of its propagation depends on the wavelength v m \u003d a νm. (1.60) It can be seen from this formula that waves of different lengths propagate with different speeds (waves disperse). The velocity of propagation of short-wave disturbances (large m) undergoes more significant changes. The term νu xxx with the third derivative of the solution is responsible for the wave dispersion. 18

20 Having considered the behavior of individual harmonics, we can now predict the qualitative behavior of the solution (1.55) of the Cauchy problem for these equations. For example, let the initial function u (x, 0) have the form of a step (1, x 0, u (x, 0) \u003d (1.61) 0, x\u003e 0 and a\u003e 0. The expansion of such a function in the Fourier series (1.54) will contain the entire set of harmonics.The solution of the Cauchy problem for the transport equation (1.51) is represented in the following form u (x, t) \u003d mbme im (x at) \u003d mbme imξ \u003d u (ξ, 0), (1.6) that is, the solution to the problem is the initial profile moving with velocity A. Solution u (x, t) \u003d mbme µmt e im (x at) \u003d mbme µmt e imξ (1.63) to the Cauchy problem for equation (1.5) with a dissipative term in which short waves are strong decay, will have the form of a smeared step.Finally, the solution u (x, t) \u003d mbme im (x (a νm) t) (1.64) to the Cauchy problem for equation (1.53), in which waves of different lengths move at different speeds, has non-monotonic, oscillating. According to formula (1.60), for ν\u003e 0, waves of short length will have a velocity less than waves of long length, and for ν< 0 наоборот. Поэтому осцилляции будут отставать от основного решения (описываемого первыми гармониками) при ν > 0 and, accordingly, move ahead at ν< Дифференциальное приближение разностной схемы. Вернемся к численному решению задачи Коши для уравнения переноса (1.51). В качестве начального профиля возьмем ступеньку { 1, x x0, u(x, 0) = (1.65) 0, x > x 0 19

21 and carry out the calculation according to the explicit upwind scheme un + a un un 1 \u003d 0, a \u003d const\u003e 0. (1.66) As a result, we obtain a solution in the form of a smeared step (Fig. 3), that is, the solution will be qualitatively the same as and a solution to equation (1.5) with a dissipative term. What's the matter? After all, we wanted to solve the transport equation in which there is no dissipative term. The point is that we were looking for a numerical solution not to the transport equation, but a solution to the difference scheme. Thus, the properties of solutions of the approximated differential equation and the approximating difference equation may not coincide. How, then, can we predict the properties of the solution to the difference equation? y x 30 Fig. 3. Graphs of the exact solution (dashed lines) and the numerical solution (solid lines) obtained using the upwind scheme (1.66) at times t \u003d 1 (1); t \u003d 8 (); t \u003d 15 (3). a \u003d 1; x 0 \u003d 10; a / \u003d 0, 5 This can be done using the differential approximation method, which we will now briefly acquaint ourselves with. The essence of this method is to replace the original difference equation with a special differential equation that has all the properties of the difference equation under study. Therefore, instead of investigating the difference equation, this differential equation is investigated, which in many cases is much easier to do. Obtaining a differential equation corresponding to a difference equation begins with writing this difference equation in the form of a so-called theoretical difference scheme, in which the difference operators act in the same functional space as the differential operators they approximate. For example, the difference equation (1.66) is written as the following theoretical difference 0

22 circuits u (x, t +) u (x, t) u (x, t) u (x, t) + a \u003d 0. (1.67) The solution of such a circuit is the function u (x, t) of continuous arguments x and t while the solution to equation (1.66) is the grid function u defined only at the grid nodes. Let a sufficiently smooth function u (x, t) be a solution to the theoretical difference scheme (1.67). Let us substitute it into this scheme and express u (x, t +) and u (x, t) in terms of the values \u200b\u200bof the function and its derivatives at the point (x, t) by the Taylor formula. As a result, we obtain a differential equation equivalent to the difference scheme (1.67) u t + au x + u tt + 6 u ttt a u xx + a 6 u xxx + ... \u003d 0. (1.68) Definition. The differential equation of infinite order (1.68) obtained after the expansion according to the Taylor formula of the solution u (x, t) of the theoretical difference scheme (1.67) is called the differential representation of the difference scheme (1.66). Some properties of the difference scheme can be investigated already with the help of this differential representation, but for our purposes it will be more convenient to use a different form of the differential representation resulting from the elimination from (1.68) of all time derivatives except for the one included in the approximated equation (1.51), i.e. that is, except for u t. Let us show, for example, how to eliminate time derivatives in terms of order and. For this, we rewrite equation (1.68) taking into account the terms up to the order O () and O () ut + au x + u tt + 6 u ttt au xx + a 6 u xxx \u003d O () (1.69) and find using the resulting equation derivative ut: ut \u003d au xu tt 6 u ttt + au xx a 6 u xxx + O () (1.70) We substitute this derivative into the terms of equation (1.69) containing the derivatives (ut) t and (ut) tt. Taking into account the order of smallness of the coefficients of the second and third time derivatives, we obtain that in (u t) t 1

23, it is sufficient to substitute the derivative (1.70) calculated with an accuracy of O (+): ut \u003d au xu tt + au xx + O (+), (1.71) and in (ut) tt with an accuracy of O (+): ut \u003d au x + O (+). (1.7) As a result of this substitution, equation (1.69) takes the following form: ut + au x + (au xu tt + a) u xx + t 6 (au x) tt \u003d \u003d au xx a 6 u xxx + O (), or ut + au xau tx 4 u ttt + a 4 u txx a 6 u ttx \u003d \u003d au xx a 6 u xxx + O (). (1.73) Having made substitutions into equation (1.69), then we take similar actions with equation (1.73). Now we need to substitute the derivative ut, determined from equation (1.73), into the four terms of the same equation: ut + au xa (au x + au tx + au xx) x 4 (au x) tt + + a 4 (au x) xx a 6 (au x) tx \u003d au xx a 6 u xxx + O (). After reducing similar ones, we obtain the equation ut + au xa 1 u txx + a 4 u ttx \u003d \u003d a (a) (1 r) u xx + au xxx + O (), 6 (1.74) in which, in contrast to (1.69) , there are no second time derivatives. We calculate the mixed derivatives u txx and u ttx remaining in (1.74) based on equality (1.7): u txx \u003d au xxx + O (+), u ttx \u003d a u xxx + O (+). (1.75)

Therefore, the differential representation (1.74) takes the form u t + au x \u003d a (1 r) u xx a 6 (r 3r + 1) u xxx + O (). (1.76) Thus, we got rid of time derivatives at powers and. But the derivatives with respect to t still remain at higher powers on the right side of O (). If we continue the described procedure further, then in the representation (1.68) we can remove the time derivatives up to an arbitrarily high order. As a result, we obtain a differential representation of the circuit in the form either ut + au x \u003d a (1 r) u xx + a 6 (1 r) (r 1) u xxx + ... (1.77) ut + au x \u003d k \u003d ckkux k ... (1.78) Definition. The equation of infinite order (1.78) is called the P-form of the differential representation of the difference scheme. Let the difference scheme have orders of approximation γ 1 and γ with respect to and, respectively. Definition. The differential equation obtained from the P-form of the differential representation by discarding terms of order O (γ1 + 1, γ + 1) and higher is called the first differential approximation (p.d.p) of the difference scheme. For the upwind scheme (1.66), the p.d.p. is a second-order differential equation ut + au x \u003d µu xx, µ \u003d a (1 r), (1.79) which, as we see, coincides with Eq. (1.5) with the dissipative term ... Thus, for r 1, our scheme implicitly introduces viscosity (dissipation) into the approximated transport equation, which is called the approximation or scheme viscosity. The presence of approximate viscosity leads to smearing of the initial step. Definition. The property of a difference scheme due to the presence of derivatives of even order in its p.d.p. is called numerical dissipation. 3

25 The P-form of the differential representation of the Lax-Wendroff difference scheme has the form ut + au x \u003d a 6 (1 r) u xxx a3 8 r (1 r) u xxxx + ..., and the p.d.p. ut + au x + νu xxx \u003d 0, ν \u003d a 6 (1 r) (1.80) coincides with equation (1.53) with a dispersion term. Consequently, for r 1, the Lax – Wendroff scheme implicitly introduces dispersion into the approximated transport equation, so the solution of the difference scheme can oscillate (Fig. 4). y Fig. 4. Plots of the exact solution (dashed lines) and the numerical solution (solid lines) obtained using the Lax-Wendroff scheme at times t \u003d 1 (1); t \u003d 8 (); t \u003d 15 (3). a \u003d 1; x 0 \u003d 10; a / \u003d 0, 5 Definition. The property of a difference scheme due to the presence of odd-order derivatives in its p.d.p. is called the numerical variance. Let's summarize our reasoning. For problems with a smoothly varying solution, the contribution of high-frequency harmonics to which is small, the accuracy of the Lax-Wendroff scheme is higher than the accuracy of the upstream scheme. If we solve numerically a problem in which the solution has a sharply varying monotonic profile, then the use of a first-order upwind scheme will give a monotonic non-oscillating profile, but strongly smoothed. This is the result of numerical dissipation. The Lax - Wendroff scheme with numerical variance can give nonmonotonic profiles of the numerical solution in the vicinity of a discontinuity or a sharp change in the solution, distorted by nonphysical oscillations. x 4

26 CHAPTER 1.1. Show that for a< 0 схема (1.8) абсолютно неустойчива. 1.. С помощью спектрального метода Неймана показать, что явная схема для уравнения (1.1) u n + a un +1 un = 0, n = 0,..., M 1, = 0, ±1, ±,... (1.81) при a > 0 is absolutely unstable Using the Neumann spectral method, derive the necessary stability condition for the three-layer leap-frog scheme (leapfrog scheme, leapfrog scheme) for equation (1.1) un 1 + a un +1 un 1 \u003d 0, n \u003d 1, ..., M 1, \u003d 0, ± 1, ±, ..., (1.8) if the passage to the limit is given in the form (1.33) Determine the order of approximation of an explicit scheme with central difference un + a un +1 un 1 \u003d 0 , (1.83) constructed for the transport equation (1.1). Using the Neumann spectral method, investigate the stability of this scheme if the law to the limit is given in the form a \u003d const. (1.84) 1.5. Determine the order of approximation of the majorant scheme u n + a un +1 un 1 \u003d a un +1 un + un 1, (1.85) constructed for the transport equation (1.1). Using the Neumann spectral method, investigate the stability of this scheme if the passage to the limit is given in the form (1.84). five

27 1.6. Determine the order of approximation of the McCormack scheme u un + a un +1 un \u003d 0, 0, 5 (u +) un / + a u u 1 \u003d 0, (1.86) constructed for the transport equation (1.1). Using the Neumann spectral method, investigate the stability of this scheme if the passage to the limit is given in the form (1.84) Determine the order of approximation of the upwind scheme with weights un + σa un (1 σ) a un un 1 \u003d 0, (1.87) constructed for the transport equation ( 1.1) with a coefficient a\u003e 0. Using the Neumann spectral method, derive the necessary stability condition for scheme (1.87) if the law to the limit is given in the form (1.84) Using the maximum principle, investigate the stability in the uniform norm of the implicit upstream scheme un + a un + 1 1 \u003d f n + 1, \u003d 1, ..., N, un 0 \u003d µ n 0, n \u003d 0, ..., M, u 0 \u003d u 0 (x), \u003d 0, ..., N , (1.88) constructed for problem (1.7) Using the maximum principle, find a sufficient condition for stability in the uniform norm of an upwind scheme with weights un + σa un (1 σ) a un un 1 \u003d f n + 1 /, un 0 \u003d μ n 0 , n \u003d 0, ..., M, u 0 \u003d u 0 (x), \u003d 0, ..., N, (1.89) constructed for problem (1.7). Here 0 σ 1.6

28 1.10. Using the maximum principle, prove that the fulfillment of condition (1.44) is sufficient for the stability of the upwind scheme (1.41) with a variable coefficient a (x, t) Obtain p.d.p. (1.80) of the Lax-Wendroff scheme.Find the d.s.s. implicit the scheme un + a un + 1 1 \u003d 0, (1.90) constructed for the transport equation (1.1) with a coefficient a\u003e 0. Give a qualitative explanation of the behavior of the solution of the difference scheme for t\u003e 0, if at the initial time t \u003d 0 the step ( 1.61) .. The property of monotonicity of difference schemes. 1. One of the main requirements for difference schemes is that the solution of the difference equation should convey the features of the behavior of the solution of the approximated differential equation. Consider, for example, the Cauchy problem for the linear transport equation u t + au x \u003d 0, a \u003d const\u003e 0,< x <, t > 0, (.1) u (x, 0) \u003d u 0 (x). (.) If u 0 (x) is a non-decreasing (non-increasing) function of the variable x, then for any fixed t\u003e 0 the solution u (x, t) of the problem (.1), (.) Will also be a non-decreasing (non-increasing) function of the variable x. This follows from the fact that at any moment of time the solution is given by the formula u (x, t) \u003d u 0 (x at). (.3) It is natural to require that the solution of the difference scheme approximating problem (.1), (.) Also has a similar property. But it turns out that many difference schemes violate the monotonicity of the numerical solution: instead of the expected monotonic profiles, solutions containing nonphysical oscillations are obtained (Fig. 4). The reason for their occurrence is the numerical variance of the difference 7

29 circuits discussed in the previous paragraph. In this section, we present the conditions under which the difference scheme will maintain the monotonicity of the numerical solution. Consider an arbitrary explicit difference scheme \u003d α b α u n + α, (.4) where α is an integer, α \u003d α 1, α 1 + 1, ..., α, nodes x + α define the scheme template. Definition. A difference scheme (.4) is called a scheme that preserves the monotonicity of the numerical solution (monotonic scheme) if it transforms any monotonic function u n into a function monotonic on the (n + 1) th time layer, and with the same growth direction. Example 1. Let us approximate equation (.1) on a uniform grid by the upwind scheme u n + a un un 1 \u003d 0. (.5) This scheme has the first order of approximation in and. Let the grid function u n on the n-th time layer be monotonic, for example, a monotonically increasing function, that is, u n un 1 for arbitrary. In this case, under the condition of stability of the circuit, which has the form aæ 1, where æ \u003d /, we obtain 1 \u003d (un aæ (unun 1)) (un 1 aæ (un 1 un)) \u003d (1 aæ) (unun 1) + aæ (un 1 un) 0. Thus, the solution monotonically increases on the (n + 1) th layer. Thus, the upwind scheme (which dissipates at a< 1) является схемой, сохраняющей монотонность. Пример.. Покажем, что схема Лакса Вендроффа (1.49) (не обладающая диссипацией при aæ < 1) не сохраняет монотонность численного решения. Пусть начальная функция для уравнения (.1) имеет вид (1.61) { 1, при x 0, u 0 (x) = 0, при x > 0. 8

30 Therefore, the initial grid function (u 0 1, for 0, \u003d u 0 (x) \u003d 0, for\u003e 0 is monotonically decreasing. We rewrite the considered scheme in the form of a one-step scheme (1.50), and then in the form of scheme (.4) with coefficients b 1 \u003d a æ + aæ, b 0 \u003d 1 a æ, b 1 \u003d a æ aæ. (.6) Then it is easy to verify that on the first layer in time equality 1 holds, for 1, u 1 b \u003d 1 + b 0, for \u003d 0, b 1, for \u003d 1, 0, for. For aæ< 1 схема устойчива, но b 1 + b 0 > 1, that is, the grid function u 1 is not monotonically decreasing. The monotonicity of schemes for equations with constant coefficients can be investigated using the following theorem. Theorem. 1. In order for the difference scheme (.4) with constant coefficients b α to remain monotonic, it is necessary and sufficient that the conditions b α 0 be satisfied for all α. (.7) PROOF. Need. Suppose that the scheme (.4) is monotonic, but there is a negative coefficient b α0< 0. Возьмем монотонно возрастающую функцию u n = { 0, < α0, 1, α 0. (.8) Тогда u0 n+1 1 = b α u n α b α u n 1+α = α α = b α b α = b α0 < 0, α α 0 α α

31 that is, the function is not monotonically increasing, and, therefore, scheme (.4) does not preserve monotonicity, which contradicts the initial assumption. This contradiction proves that all coefficients b α are nonnegative. Adequacy. Let b α 0 and u n be a monotone function, for example, a monotonically increasing function. Then 1 \u003d α b α u n + α α b α u n 1 + α \u003d \u003d α b α (u n + α u n 1 + α) 0, that is, it is also a monotonically increasing function. Thus, scheme (.4) is monotonic. Let us return again to examples 1 and., And now we will not assume that a\u003e 0. The counterflow scheme for equation (.1) with an arbitrary sign of the coefficient a looks like this: where un We rewrite it in the form (.4) + a + un un 1 a + \u003d a + a + a un +1 un, a \u003d a a. \u003d 0, (.9) where \u003d b 1 u n 1 + b 0 u n + b 1 u n +1, (.10) b 1 \u003d æa +, b 0 \u003d 1 æ a, b 1 \u003d æa. Under the stability condition a æ 1 (.11), all these coefficients are non-negative. Moreover, they are constant, therefore, according to Theorem 1, the upwind scheme (.9) preserves the monotonicity of the solution under condition (.11). The Lax - Wendroff scheme is stable under the same condition (.11) as the upwind scheme, and it can be written in the form (.10) with coefficients (.6), whence it can be seen that under the condition a æ< 1 один из 30

32 coefficients b 1 or b 1 are negative. According to Theorem 1, this implies that the Lax - Wendroff scheme, which has the second order of approximation in and, does not preserve the monotonicity of the numerical solution. But, perhaps, there are other schemes of the second order of approximation that have the property of monotonicity. It turns out that there are no such schemes. It is shown in the paper that for the linear transport equation (.1) it is impossible to construct a monotone scheme with constant coefficients of the second order of approximation ... Now consider the scheme (.4) with variable coefficients b α. Will the condition (.7) of nonnegativity of the coefficients for such schemes be sufficient to preserve the monotonicity of the numerical solution? It turns out not. Let's give an example. Example 3. Let the Cauchy problem be solved for the equation u t + a (x) u x \u003d 0, (.1) where a (x) is a strictly increasing positive bounded function: 0< a(x) < 1 и a > 0. To solve this problem, we take a scheme with variable coefficients 0, 5 (u n +1 +) un 1 + a u n +1 un 1 \u003d 0, (.13) where a \u003d a (x), x is a node of a uniform grid. The described scheme is an analogue of the Lax scheme (1.46), which preserves the monotonicity of the numerical solution (see Problem 1). We will assume that the condition a< 1, (.14) гарантирующее устойчивость схемы (.13) в равномерной норме по начальным данным: C u 0 C. (.15) Запишем схему (.13) в виде (.4): = b 1, u n 1 + b 1, u n +1, (.16) где b 1, = 1 + æa, b 1, = 1 æa, 31

33 in this case, the coefficients b α are provided with an additional index, since they are variable coefficients and change when passing from one node to another. By virtue of condition (.14), both coefficients are positive, but scheme (.13) does not preserve the monotonicity of the numerical solution. Indeed, taking a monotonically increasing function (u n 0,< 0, = 1, 0, убеждаемся, что на (n + 1)-м слое по времени имеет место равенство = 0, при < 1, b 1, 1, при = 1, b 1,0, при = 0, 1, при 1. Но b 1, 1 > b 1.0, therefore the grid function is increasing. is not monotonic The above example shows that for schemes with variable coefficients, other monotonicity criteria should be used than the indicator (.7) indicated in Theorem.1. Theorem .. Let the coefficients of the difference scheme \u003d b 1, un 1 + b 0, un + b 1, un +1 (.17) satisfy the condition at each node x Then the fulfillment for all conditions b 1, + b 0, + b 1 , \u003d 1. (.18) b ± 1, 0, b 1, + b 1, 1 1 (.19) is necessary and sufficient for the scheme (.17) with variable coefficients to preserve the monotonicity of the numerical solution. PROOF. Let us write the scheme (.17) with variable coefficients satisfying condition (.18) in the following form: \u003d u n b 1, (u n u n 1) + b1, (u n +1 u n). (.0) 3

34 Then +1 \u003d un +1 b 1, + 1 (u n +1 u n) + b1, + 1 (u n + u n +1). Therefore, +1 un + 1 \u003d (un +1 un) (1 b 1, + 1 b 1,) + (+ b 1, + 1 un + un (+1) + b 1, unun) (.1) 1. Necessity. Let the scheme (.17) be monotonic. Let us prove that its coefficients satisfy inequalities (.19). Suppose that this is not the case and some of the conditions (.19) are not satisfied at some node x 0, for example b 1,0< 0. Положим Из (.1) тогда следует u n = { 0, если < 0, 1, если un+1 0 = b 1,0 < 0, т. е. функция не является монотонно возрастающей, что противоречит исходному предположению о монотонности схемы (.17). Аналогично проверяются и остальные неравенства в (.19). Достаточность. Пусть в каждом узле x коэффициенты схемы (.17) удовлетворяют неравенствам (.19) и функция u n является монотонной, например монотонно возрастающей. Тогда из равенства (.1) следует, что функция также будет монотонно возрастающей функцией. Теорема доказана. Нетрудно проверить, что коэффициенты схемы (.16) не удовлетворяют второму из условий (.19) теоремы.3, поэтому эта схема не является схемой, сохраняющей монотонность численного решения. Дадим другую формулировку теоремы.. Теорема.3. Для того чтобы конечно-разностная схема u n + C 1/ un x, 1/ C+ +1/ un x,+1/ = 0, (.) сохраняла монотонность численного решения, необходимо и достаточно выполнение при всех условий где æ = /, u n x,+1/ = un +1 un C ± +1/ 0, C +1/ + C+ +1/ 1 æ, (.3). 33

35 PROOF. Scheme (.) Can be rewritten in the form (.17), while b 1, \u003d æc 1 /, b 1, \u003d æc + + 1 /, b 0, \u003d 1 æc 1 / æc + + 1 /. Then for the coefficients b α equality (.18) holds, and conditions (.19) are equivalent to conditions (.3). Comment. It is proved in the work that the fulfillment of inequalities (.3) is sufficient for the scheme (.) To be a TVD-scheme (Total Variation Diminising Sceme), i.e., a scheme whose solution un for any n 0 satisfies the condition of non-increase of the total variation TV () TV (un), (.4) where by the total variation of the grid function un we mean the value TV (un) \u003d un +1 u n. (.5) Currently, TVD schemes and their various modifications are used to solve many problems with discontinuous solutions. The reason for such a great popularity of these methods is that they give non-oscillating profiles of the solution, high resolvability in the discontinuity region, and retain high accuracy in the regions of smoothness of the solution. Modern TVD schemes of a high order of approximation are based on various methods of restoring (reconstructing) the values \u200b\u200bof functions at the boundaries of cells from their values \u200b\u200bat the centers of neighboring cells. In this case, the schema template is variable and depends on the behavior of the numerical solution. Reconstruction algorithms are based on the use of special flow limiters, which are constructed so that the circuit with limiters has the TVD property (.4) .. 3. Monotonization of the Lax-Wendroff scheme. If the initial function at t \u003d 0 is given in the form of a step, then on the next layers in time we will obtain a step distorted by oscillations according to the Lax-Wendroff scheme (see Fig. 4). But it turns out that the Lax Wendroff scheme can be modified so that it has 34

36 TVD property (.4), and hence, according to Theorem 3, would become a scheme that preserves the monotonicity of the numerical solution. However, the coefficients of the modified circuit will no longer be constant, they may depend on the solution at the n-th layer, i.e., the modified circuit will be nonlinear. Consider the transport equation (.1) in the case a \u003d const\u003e 0. The Lax - Wendroff scheme (1.50) can be rewritten as un + a un x, + 1 / + un x, 1 / a () unx, + 1 / un x, 1 / \u003d 0, (.6) or un + au nx, 1 / + a (1 aæ) un x, + 1 / un x, 1 / un \u003d 0, (.7) + au nx, \u003d a (1 aæ) un xx ,. (.8) The exponential function (1.79) of the upstream scheme contains the dissipative term 0, 5a (1 aæ) u xx on the right-hand side, and in the representation (.8) the same dissipative term in the difference form has the opposite sign. Thus, the Lax-Wendroff scheme is represented as a monotonic scheme with a difference directed upstream, supplemented by the so-called anti-diffusion term, which eliminates the dissipative term in the pd.p. of the upstream scheme, transforming it into the Lax-Wendroff scheme. By reducing the anti-diffusion term in the places where oscillations may appear, one can try to prevent them. We will regulate the anti-diffusion term in the Lax-Wendroff scheme (.7) using the limiting function Φ (ξ) of some argument ξ: un + au nx, 1 / + a (1 aæ) ((Φu nx) + 1 / (Φu nx) 1 /) \u003d 0. (.9) If Φ 0, then we have a monotone upwind scheme of the first order of approximation. If Φ 1, then we obtain the Lax - Wendroff scheme of the second order of approximation on smooth solutions, but oscillating on discontinuous solutions. 35

37 In the difference scheme (.9) Φ + 1 / \u003d Φ (ξ + 1 /). As a discrete argument ξ + 1 / we choose the quantity unx, 1 / ξ + 1 / \u003d un for unx, + 1/0, x, + 1 / (.30) 1 for unx, + 1 / \u003d 0. On the oscillating solution the ratio unx, 1 / / un x, + 1 / becomes negative, therefore, for ξ + 1 /< 0 полагаем, что Φ +1/ = 0. Далее будем считать, что функция Φ = Φ(ξ) непрерывного аргумента ξ также принимает нулевые значения при ξ < 0. Более того, предполагая, что функция-ограничитель является непрерывной, полагаем, что Φ(ξ) 0 при всех ξ 0. Далее рассмотрим случай, когда ξ +1/ > 0. We will select the limiter function so that the circuit satisfies the TVD condition (.3) and retains the second order of approximation on smooth solutions. For this, we transform the modified Lax-Wendroff scheme (.9) to the form (.): Either un + au nx, 1 / + a (1 aæ) ((Φ ξ un [+ a aæ ((Φ) ξ) + 1 / + 1 / Φ 1 /) unx, 1 / \u003d 0, Φ 1 /)] unx, 1 / \u003d 0. Thus, the coefficients of the scheme (.9) written in the form (.) Are determined by the formulas [C + +1 / \u003d 0, C 1 / \u003d a aæ ((Φ))] ξ Φ 1 /. + 1 / According to Theorem 3, the condition 0 C 1/1 æ (.31) will guarantee that the Lax - Wendroff scheme with the introduced limiter function will preserve the monotonicity of the numerical solution. In what follows, we assume that the stability condition for the Lac-36

38 sa Wendroff holds, that is, aæ 1. Then, for inequalities (.31) to be valid for all aæ 1, it is necessary and sufficient that the inequalities (Φ) ξ + 1 / Φ 1 / hold, and for this it is sufficient to require for all the following inequalities: (Φ) 0, 0 Φ + 1 /. ξ + 1 / The region in the plane of the variables Φ and ξ, in which these inequalities hold, is shown in Fig. 5, a. If the graph of the function Φ \u003d Φ (ξ) lies in this region, then the modified scheme (.9) will preserve the monotonicity of the numerical solution. Φ Φ \u003d Φ Φ \u003d Φ \u003d ξ Φ \u003d ξ Φ \u003d ξ 1 1 Φ \u003d a ξ b ξ Fig. 5. and in the shaded area, the modified Lax Wendroff scheme (.9) is a TVD scheme; b in the double shaded domain the modified Lax - Wendroff scheme is a TVD scheme of the second order of approximation So, in what follows we will assume that Φ (ξ) \u003d 0 for ξ 0, 0 Φ (ξ) min (, ξ) for ξ\u003e 0. ( .3) Let us now investigate the order of approximation of the modified scheme, assuming that the continuous function Φ \u003d Φ (ξ) satisfies 37

39 with the following additional constraints: Φ (ξ 1) Φ (ξ) L ξ 1 ξ, ξ 1, ξ, (.33) Φ (1) \u003d 1, (.34) i.e., we require that the function Φ \u003d Φ (ξ) satisfied the Lipschitz condition with some constant L\u003e 0 and the graph of this function passed through the point (1, 1). We rewrite the modified Lax-Wendroff scheme (.9) in the form of the original Lax-Wendroff scheme (.7) with an additional term where un + au nx, 1 / + a (1 aæ) (unx, + 1 / un x, 1 /) + + a (1 aæ) Rn \u003d 0, (.35) R n \u003d (Φ + 1/1) unx, + 1 / (Φ 1/1) unx, 1 /. (.36) Let u \u003d u (x, t) be a sufficiently smooth solution to the Cauchy problem (.1), (.). Let us substitute this solution into expression (.36), keeping all the previous notation in it, but taking into account that now u n x, + 1 / \u003d u (x +1, t n) u (x, t n). (.37) Obviously, if on the nth layer in time the function u (x, tn) is linear, u (x, tn) \u003d Bx + C, then R n 0. Using conditions (.33), (. 34), it is easy to check that for a quadratic function u (x, tn) \u003d Ax + Bx + C (A 0) the equality R n \u003d O () holds for all nodes of an arbitrary numerical interval (α, β) not containing the point extremum x \u003d B / A. In the general case, the following statement is true. Lemma. 1. Let conditions (.33), (.34) be satisfied and a sufficiently smooth solution to the Cauchy problem (.1), (.) Satisfies on some numerical interval [α, β] the condition ux (x, tn) 0 x [α, β] ... (.38) Then R n \u003d O () x (α, β). (.39) 38


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Chapter 6 Foundations of stability theory Lecture Statement of the problem Basic concepts Earlier it was shown that the solution of the Cauchy problem for a normal system ODE \u003d f, () continuously depends on the initial conditions at

Chapter 9. Numerical methods. Lecture 4. Euler's difference method for solving the Cauchy problem for differential equations .. Euler's differential and difference problem. Definition. Euler's differential problem

DIFFERENTIAL EQUATIONS General concepts Differential equations have numerous and most diverse applications in mechanics, physics, astronomy, technology and in other branches of higher mathematics (for example

Linear and nonlinear equations of physics Laplace's equation in a polar coordinate system. Senior Lecturer of the Department of the Navy Levchenko Evgeniy Anatolievich 518 Chapter 5. Equations of elliptic type 25.2. Separation

Lecture 3 Stability of equilibrium and motion of the system When considering steady motions, we write the equations of perturbed motion in the form d dt A Y where the column vector is the square matrix of constant coefficients

Numerical series Numerical sequence Opr A numerical sequence is a numerical function defined on the set of natural numbers x - a common term of the sequence x \u003d, x \u003d, x \u003d, x \u003d,

5 Power series 5 Power series: definition, domain of convergence Functional series of the form (a + a) + a () + K + a () + K a) (, (5) where, a, a, K, a, k are some numbers, called power series Numbers

Ministry of Education of the Russian Federation MATI - RUSSIAN STATE TECHNOLOGICAL UNIVERSITY named after KE TSIOLKOVSKY Department of Higher Mathematics VV Gorbatsevich K Yu Osipenko Partial Equations

Let us now consider the simplest difference schemes for the Hopf equation.

The generalization of the Lax scheme to the case of the Hopf equation has the form

Here, obviously, the divergent form of equation (3.6) is used.

Exercises... Consider the Lax - Wendroff scheme for the Hopf equation. Let the initial conditions for the Cauchy problem be formulated as follows: u (x, 0) \u003d ch - 2 (x). Then the Hopf equation has the first integral: ... Check that the above diagram is conservative, i.e. the same conservation law is automatically fulfilled in it at the grid level.

Build a similar circuit using characteristic form writing the Hopf equation (3.9). Will she be conservative?

The scheme is conditionally stable under the Courant condition (more precisely, a generalization of the Courant condition)

Here and below, as before in (3.7), f \u003d 0.5u 2. In this case, it is assumed that the flow is sufficiently smooth, the moment of the gradient catastrophe has not yet arrived, and there are no shock waves and other discontinuities in the solution.

Courant - Isakson - Rhys scheme... Generalization of the KIR schemes to the quasilinear case (when using divergent form equations) is obvious.

The scheme is stable under the Courant condition

Generalization lax-Wendroff schemes (predictor-corrector scheme). For quasilinear equations (as well as linear equations with variable coefficients, inhomogeneous equations, etc.), the Lax - Wendroff scheme becomes more complicated. To construct it, it is necessary to introduce the so-called half-integer points (points with fractional indices). At the first stage (predictor), the values \u200b\u200bat half-integer points are calculated according to the above scheme - a generalization to the quasilinear case of the Lax scheme:

at the second stage (corrector), the "leapfrog" scheme is used (a three-layer scheme on a cruciform pattern, which is not included in the family (3.8)):

The Lax - Wendroff scheme belongs to the so-called central schemes. Its pattern is symmetrical. At the first stage, the values \u200b\u200bof the grid function are calculated at half-integer points of the template on the intermediate layer (tm - 1/2, xm - 1/2), (tn + 1/2, xm + 1/2), at the second stage, the solution on the upper layer is calculated at the point (tn + 1, xm). The scheme is stable under the Courant condition.

Lax - Wendroff schemes for linear inhomogeneous equations are constructed in a similar way.

McCormac's off-center circuit (predictor - corrector).

Like the Lax-Wendroff scheme above, the McCormack scheme consists of two stages. Consider the construction of the McCormack circuit for homogeneous equation (3.7). The first stage (predictor) has the form

those. the explicit right corner scheme is used. The second stage is a proofreader:

Thus, the calculation at the first stage according to the "right corner" scheme, at the second - "left corner".

Another McCormack scheme has the form

Such difference schemes are called off-center... Their advantages include the absence of half-integer indices, and a simpler formulation of boundary conditions. In the linear case, the McCormack schemes coincide with the Lax-Wendroff scheme. The schemes have the second order of approximation in both variables, the schemes are stable under the Courant condition.

Rusanov's scheme (central scheme of the third order of accuracy).

To construct the Rusanov scheme, not only half-integer points are introduced, but also two layers of intermediate points with fractional indices. The first stage of the Rusanov scheme (transition to layer 1/3) has the form

its second stage is a "leapfrog" scheme

and the third stage

At the first stage, the calculation is made according to the Lax scheme, at the second - according to the "cross" ("leapfrog") scheme. The last term of the third stage is introduced to ensure the stability of the scheme (a term proportional to the difference approximation of the 4th derivative).

The scheme is conditionally stable under the Courant condition and the condition.

Off-center warming - Kutler - Lomax scheme 3rd order of accuracy.

First stage:

Second phase:

Third stage:

The last term is added for the stability of the scheme, which is conditionally stable under the Courant conditions.