Abstract.In this paper we prove strict stability of high-order finite difference approximations of parabolic and symmetric hyperbolic systems of partial differential equations on bounded, curvilinear domains in two space dimensions. The boundary need not be smooth. We also show how to derive strict stability estimates for inhomogeneous boundary conditions. Strict stability in several space dimensionsIn [2] we proved stability for high-order finite difference approximations of hyperbolic and parabolic systems using certain projections and difference operators satisfying a summation-by-parts rule. In one space dimension we showed how to obtain strict stability. The aim of this paper is to prove strict stability in several space dimensions. Furthermore, it will also be demonstrated how to handle inhomogeneous boundary conditions. We limit ourselves to two space dimensions for convenience.The purpose of strict stability is to ensure the same growth rate of the discrete and analytic solutions. If the analytic problem is defined on a curvilinear domain Q with boundary Y (cf. Fig. 1 on next page), then there must exist a diffeomorphism Ç = Ç(x) of Q, onto the unit square (0, 1) x (0, 1) in order for the finite difference method to be well defined. Consequently, a constantcoefficient problem in the original domain may be transformed to a variablecoefficient problem on the unit square, which may account for a nonphysical growth in the discrete estimate.

We have derived stability results for high-order finite difference approximations of mixed hyperbolic-parabolic initial-boundary value problems (IBVP). The results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection. By rearranging the analytic equations slightly, we can prove strict stability for hyperbolic-parabolic IBVP. Furthermore, we generalize our technique so as to yield stability on nonsmooth domains in two space dimensions. Using the same procedure, one can prove stability in higher dimensions as well.Proposition 2.1. There exist scalar products (2.2) and difference operators D of accuracy 2p -1 at the boundaries and 2p in the interior, p > 0, such that the summation-by-parts property (2.1) holds.Confining ourselves to the case where Z(1) and S(2) are diagonal, we have the following existence theorem [4].Proposition 2.2. There exist diagonal scalar products (2.2) and difference operators D of accuracy p at the boundaries and 2p in the interior, 1 < p < 4, such that the summation-by-parts property (2.1) holds.

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Abstract.In this paper we prove strict stability of high-order finite difference approximations of parabolic and symmetric hyperbolic systems of partial differential equations on bounded, curvilinear domains in two space dimensions. The boundary need not be smooth. We also show how to derive strict stability estimates for inhomogeneous boundary conditions. Strict stability in several space dimensionsIn [2] we proved stability for high-order finite difference approximations of hyperbolic and parabolic systems using certain projections and difference operators satisfying a summation-by-parts rule. In one space dimension we showed how to obtain strict stability. The aim of this paper is to prove strict stability in several space dimensions. Furthermore, it will also be demonstrated how to handle inhomogeneous boundary conditions. We limit ourselves to two space dimensions for convenience.The purpose of strict stability is to ensure the same growth rate of the discrete and analytic solutions. If the analytic problem is defined on a curvilinear domain Q with boundary Y (cf. Fig. 1 on next page), then there must exist a diffeomorphism Ç = Ç(x) of Q, onto the unit square (0, 1) x (0, 1) in order for the finite difference method to be well defined. Consequently, a constantcoefficient problem in the original domain may be transformed to a variablecoefficient problem on the unit square, which may account for a nonphysical growth in the discrete estimate.

In this paper In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave.In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method.The results show that the fourth order methods are the only ones that give good results fur all the considered test problems.