Volume 1, Issue 3
A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms

S. Qamar & G. Warnecke

DOI:

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

Published online: 2006-01

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  • 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 [1]. 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 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.

  • Keywords

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

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@Article{CiCP-1-449, author = {}, title = {A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms}, journal = {Communications in Computational Physics}, year = {2006}, volume = {1}, number = {3}, pages = {449--478}, 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 [1]. 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 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.

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7964.html} }
TY - JOUR T1 - A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms JO - Communications in Computational Physics VL - 3 SP - 449 EP - 478 PY - 2006 DA - 2006/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7964.html KW - Hyperbolic systems with relaxation KW - stiff systems KW - space-time conservative and Jacobian-free method KW - high order accuracy KW - discontinuous solutions. AB -

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 [1]. 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 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.

S. Qamar & G. Warnecke. (2020). A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms. Communications in Computational Physics. 1 (3). 449-478. doi:
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