High order finite difference methods with subcell resolution for advection equations with stiff source terms

• Published in: Journal of computational physics 2012, Vol.231 (1), p.190-214
• Main Authors: Wang, Wei, Shu, Chi-Wang, Yee, H.C., Sjögreen, Björn
• Format: Article
• Language: English
• Subjects: Computation; Computational techniques; Convection; Detonation; Discontinuity; ENO subcell resolution; Exact sciences and technology; Finite difference method; Mathematical analysis; Mathematical methods in physics; Mathematical models; MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; Physics; Scalars; Shock capturing; Solvers; Stiff reaction term; WENO
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2011.08.031

► A high order WENO scheme with subcell resolution for reactive flows is proposed. ► The method is able to capture the correct discontinuity location on coarse meshes. ► The method can be applied to any high order shock capturing base scheme. ► 1D and 2D numerical examples are shown to demonstrate the effectiveness of this method. A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.