05-05
A High Order Slope Propagation Method for Hyperbolic Conservation Laws
by Ain, Q.; Qamar, S.; Warnecke,G.
Preprint series: 05-05, Preprints
The paper is published: Sumitted to the Journal
- MSC:
- 65M99 None of the above but in this section
- 35L05 Wave equation
- 35L45 Initial value problems for hyperbolic systems of first-order PDE
- 35L60 Nonlinear first-order PDE of hyperbolic type
Abstract: We present a second order scheme which treats the space and time in a unified manner for the numerical solution of hyperbolic systems. The flow variables and their slopes are the basic unknowns in the scheme. The scheme utilizes the advantages of both the CE/SE method of Chang\'s \cite{chang1} and central schemes of Nessyahu and Tadmor \cite{NT}. However, unlike the CE/SE method the present scheme is Jacobian-free and hence like the central schemes can also be applied to any hyperbolic system. By introducing a suitable limiter for the slopes of flow variables, we apply the same scheme to linear and non-linear hyperbolic systems with discontinuous initial data. However, in Chang\'s method they used a finite difference approach for the slope calculation in case of nonlinear hyperbolic equations with discontinuous initial data. The scheme is simple, efficient and has a good resolution especially at contact discontinuities. We derive the scheme for the one and two space dimensions. In two-space dimension we use triangular mesh. The second order accuracy of the scheme has been verified by numerical experiments. Several numerical test computations presented in this article validate the accuracy and robustness of the present scheme.
Keywords: Conservation laws, hyperbolic systems, space-time control volumes, finite volume schemes, high order accuracy, linear and non-linear systems, shock solutions
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