2 Multidimensional Hyperbolic Problemss where A(u) =f u (u) B(u) =g u (u): (7.1.1c) Multidimensional nite dierence schemes can be simple extensions of

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1 Chapter 7 Multidimensional Hyperbolic Problems 7.1 Split and Unsplit Dierence Methods Our study of multidimensional parabolic problems in Chapter 5 has laid most of the groundwork for our present task of creating dierence approximations for multidimensional hyperbolic problems. By now we know that denitions of consistency, convergence, and stability for one-dimensional problems carry over to multiple dimensions stability analyses by von Neumann's method are equivalent to those in one dimension but algebraically more complex and implicit schemes are dicult to implement without splitting the operator or alternating directions. Geometrical complexities will, again, be postponed until Chapter 8, so we'll consider two-dimensional conservation laws of the form u t + f(u) x + g(u) y =0 (7.1.1a) where u, f, and g are m-vectors. The convective form of this system is u t + Au x + Bu y =0 (7.1.1b) 1

2 2 Multidimensional Hyperbolic Problemss where A(u) =f u (u) B(u) =g u (u): (7.1.1c) Multidimensional nite dierence schemes can be simple extensions of one-dimensional methods. Thus, for example, a Lax-Friedrichs approximation of (7.1.1a) would be = 1 4 (Un j+1 k + U n j;1 k + U n j k+1 + U n j k;1) 2x (f n j+1 k ; f n j;1 k) 2y (gn j k+1 ; g n j k;1): (7.1.2) As in Chapter 5, a uniform mesh of spacing xy has been introduced with U n being the nite dierence approximation of u(jx ky nt). Extending the Richtmyer two-step method (6.4.1) to two dimensions is a bit more dicult. The predictor step is the Lax-Friedrichs method lm = 1 4 (Un l+1 m + U n l;1 m + U n l m+1 + U n l m;1) 2x (f n l+1 m ; f n l;1 m) 2y (gn l m+1 ; g n l m;1) (l m) =(j 1 k) (j k 1) (7.1.3a) applied to the four points surrounding (j k) (Figure 7.1.1). The corrector step is the leap frog method U n+2 = U n n+1 (fj+1 k x ; f n+1 ) j;1 k y (gn+1 ; j k+1 gn+1 ): j k;1 (7.1.3b) The computational stencil of (7.1.3) is shown in Figure The mesh spacing t, x, and y has been doubled relative to the one-dimensional scheme (6.4.1). This was done to simplify the writing of the scheme. As shown on the bottom of Figure 7.1.1, the Richtmyer two-step scheme can be regarded as a nine-point dierence formula on a staggered grid. This is possible because the physical ux vector and the divergence operator are rotationally invariant. Halving the mesh spacing to get a scheme that is more in line with our usual notation is much simpler with this staggered grid interpretation. The Courant, Friedrichs, Lewy Theorem is still available to restrict the domain of dependence of a dierence scheme to contain that of the partial dierential equation. For example, the solution of the model initial value problem u t + au x + bu y =0 ;1 <x y<1 t>0 (7.1.4a)

3 7.1. Split and Unsplit Dierence Methods 3 k j k j Figure 7.1.1: Computational stencil of the Richtmyer two-step method (7.1.3). Predicted solutions are shown with lled circles and corrected solutions are shown with blue circles and corrected solutions are shown in red. The Richtmyer two-step scheme can be regarded as a nine-point formula on a staggered grid (bottom).

4 4 Multidimensional Hyperbolic Problemss is u(x y 0) = (x y) ;1 <x<1 (7.1.4b) u(x y t) =(x ; at y ; bt): (7.1.4c) Thus, the domain of dependence of a point(x 0 y 0 t 0 ) is the single point(x 0 ;at 0 y 0 ;bt 0 ). Let us consider a ve- or nine-point explicit dierence scheme of the form = j+1 X k+1 X l=j;1 m=k;1 C lm U n lm The domain of dependence of the mesh point (j k n +1)for the nine-point scheme is the rectangle D = f(x y)jx j;1 x x j+1 y k;1 y y k+1 g: (7.1.5a) A ve-point scheme would have C j1 k1 = 0. Its domain of dependence would be less clear however, for simplicity, we'll dene it as D = f(x y)j jx ; x jj x + jy ; y kj y 1g: (7.1.5b) The domain of dependence of the point (j k n +1)for the partial dierential equation is the single point (x j ; at y k ; bt). The various domains of dependence are shown in Figure Thus, the Courant, Friedrichs, Lewy condition is max( jajt x jbjt y ) 1: (7.1.6a) for the nine-point scheme and jajt x + jbjt y ) 1 (7.1.6b) for the ve-point scheme. The Lax-Friedrichs scheme applied to (7.1.4a) yields = 1 4 (1 ; 2)U n j+1 k (1 + 2)U n j;1 k (1 ; 2)U n j k (1 + 2)U n j k;1 where = at x = bt y :

5 7.1. Split and Unsplit Dierence Methods 5 k j Figure 7.1.2: Domain of dependence of nine-point (solid line) and ve-point (dashed line) dierence schemes and of the partial dierential equation (). Using the Maximum Principle, we nd a sucient condition for stability in the maximum norm as max(jj jj) 1 2 which is more restrictive than in one dimension. Using the von Neumann method (cf. Problem 1 at the end of this section), we may show that the Lax-Friedrichs scheme is stable in L 2 when : Example With the more stringent stability restrictions, methods that use operator splitting become attractive. Let us consider splitting MacCormack's predictorcorrector scheme (6.4.2) for the conservation laws (7.1.2). In the x-direction, we neglect g, predict using forward time-backward space dierencing, and correct using backward time-forward space dierencing for one half time step to obtain ^ = U n x (f n ; f n j;1 k) (7.1.7) = Un n+1 + ^U n+1 n+1 (^f j+1 k ; ^f ) (7.1.8) 2 2x

6 6 Multidimensional Hyperbolic Problemss where ^f n+1 f( ^Un+1 ). The solution obtained at the conclusion of sweeping the mesh by rows is denoted as U n+1. The sweep in the y-direction follows the same pattern with f neglected and g included and the solution U n+1 used as initial data. Thus, ~ = U n+1 y (gn+1 ; g n+1 j k;1 ) (7.1.9) = + U ~ n+1 2 2y (~gn+1 ; j k+1 ~gn+1 ): (7.1.10) This split scheme merely needs to satisfy the one-dimensional stability conditions when the boundary conditions between the x and y sweeps are implemented with care (cf. Section 5.2). Problems 1. Analyze the stability of the Lax-Friedrichs scheme (7.1.2) for the kinematic wave equation (7.1.4) using von Neumann's method.

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