A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms (Communications in Computational Physics, 2006)
In this article we propose a higher-order space-time conservative methodfor 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 ourscheme derived for homogeneous hyperbolic systems [1]. In the present inhomogeneoussystems the relaxation time may vary from order of one to a very small value. Thesesmall values make the relaxation term stronger and highly stiff. In such situationsunderresolved numerical schemes may produce spurious numerical results. However,our present scheme has the capability to correctly capture the behavior of the physicalphenomena with high order accuracy even if the initial layer and the small relaxationtime are not numerically resolved. The scheme treats the space and time in a unifiedmanner. The flow variables and their slopes are the basic unknowns in the scheme. Thesource term is treated by its volumetric integration over the space-time control volumeand is a direct part of the overall space-time flux balance. We use two approachesfor the slope calculations of the flow variables, the first one results directly from theflux 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-freeand Riemann solver-free recipe, as well as its efficiency and high order accuracy. Inparticular 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 theliterature such as the Broadwell model, the extended thermodynamics equations, theshallow water equations, traffic flow and the Euler equations with heat transfer. Thenumerical 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.