Fourth-Order Difference Methods for Hyperbolic IBVPs

Abstract

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.

Optimal diagonal-norm SBP operators

“…To show the importance of using diagonal-norm SBP operators for this problem we present in Figure 4 the eigenvalues to (17), on a grid defined by (14) with l = 2 and m = 51. We compare the spectra using the optimal 8th order diagonal-norm SBP operator and the corresponding block-norm The accuracy properties will be tested for an analytic standing wave solution, across a discontinuous media interface (located at ξ = 0),…”

“…Deriving a strictly stable, accurate, and conservative HOFDM is a significant challenge that has received considerable past attention. (For examples, see [24,41,38,1,3,15,39,14]. )…”

Entropy Splitting and Numerical Dissipation

“…laws was established recently by Olsson and Oliger (1994) and Olsson (1995), and was applied to the two-dimensional compressible Euler equations for a perfect gas by Gerritsen and Olsson (1996) and Gerritsen (1996). Kreiss and Scherer (1977), Strand (1994) and Olsson (1995a) led to the construction of high order boundary difference operators that enabled the design of stable high order central schemes for linear hyperbolic systems.…”

“…The energy method for deriving stability estimates for hyperbolic IBVPs was first applied to the nonlinear scalar case by Gustafsson and Olsson (1994) for both higher-order central and compact symmetric schemes. It was then generalized and extended to nonlinear systems of symmetrizable hyperbolic conservation laws by Olsson and Oliger (1994) and Olsson (1995), and it was applied to the two-dimensional (2-D) compressible Euler equations for a perfect gas by Gerritsen (1996) and Gerritsen and Oisson (1996). With these recent developments, renewed interest has emerged in the use of spatial central schemes where efficiency, simplicity and non-dissipative properties are their trademarks.…”

High-order centered difference methods with sharp shock resolution

“…Yet a different kind of viscosity, based on simple 2nd-order averaging, was used in [2]. We consider the approximation of the problem in its original time dependent form…”