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2005-05

A High Order Slope Propagation Method for Hyperbolic Conservation Laws

by Ain, Q., Qamar, S., Warnecke,G..


Series: 2005-05, Preprints

MSC:
65M99 None of the above, but in this section
35L05 Wave equation
35L45 Initial value problems for first-order hyperbolic systems
35L60 Nonlinear first-order hyperbolic equations

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 v

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