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A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non Stiff Source Terms

by Qamar, S., Warnecke, G..

Series: 2005-24, Preprints

76W05 Magnetohydrodynamics and electrohydrodynamics
35L65 Conservation laws
65M06 Finite difference methods
35L45 Initial value problems for first-order hyperbolic systems
35L67 Shocks and singularities

In this article we propose a higher-order space-time conservative method
for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems. We call the scheme a
slope propagation (SP) method. It is an extension of our scheme derived for
homogeneous hyperbolic systems (Ain, Qamar and Warnecke (2005)). In the present inhomogeneous systems the relaxation time may vary from order of one to a very small
value. These small values make the relaxation term stronger and highly stiff. In such situations underresolved numerical
schemes may produce spurious numerical results. However, our present scheme has the capability to correctly capture the behavior of the physical
phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved.
The scheme treats the space and time in a unified manner. The flow variables and their slopes are the basic unknowns in the
scheme. The source term is treated by its volumetric integration over the space-time control volume and is a direct part
of the overall space-time flux balance. We use two approaches for the slope calculations of the flow variables, the first one
results directly from the
flux balance over the control volumes, while in the second one we use a finite difference approach.
The main features of the scheme are its simplicity, its Jacobian-free and
Riemann solver-free recipe, as well as its
efficiency and high of order accuracy. In particular we show that the scheme
has a discrete analog of the continuous asymptotic limit.
We have implemented our scheme for various test models available in the literature such as the Broadwell model, the extended thermodynamics
equations, the shallow water equations, traffic flow and the Euler equations with heat
The numerical results validate the accuracy, versatility and robustness of the present scheme.

Hyperbolic systems with relaxation, stiff systems, space-time conservative and Jacobian-free method, high order accuracy, discontinuous solutions