Commun. Comput. Phys., 1 (2006), pp. 449-478.


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

Shamsul Qamar 1*, Gerald Warnecke 1

1 Institute for Analysis and Numerics, Otto-von-Guericke University, PSF 4120, D-39106 Magdeburg, Germany.

Received 31 August 2005; Accepted (in revised version) 2 December 2005
Communicated by Chi-Wang Shu

Abstract

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 [Q. Ain et al. (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 transfer. The numerical results validate the accuracy, versatility and robustness of the present scheme.


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Key words: Hyperbolic systems with relaxation; stiff systems; space-time conservative and Jacobian-free method; high order accuracy; discontinuous solutions.


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Correspondence to: Shamsul Qamar , Institute for Analysis and Numerics, Otto-von-Guericke University, PSF 4120, D-39106 Magdeburg, Germany. Email: Shamsul.Qamar@Mathematik.Uni-Magdeburg.DE
† This work is supported by the Volkswagenstiftung grant I
 

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