Theoretical and numerical structure for reacting shock waves
SIAM Journal on Scientific and Statistical Computing
A study of numerical methods for hyperbolic conservation laws with stiff source terms
Journal of Computational Physics
A Modified Fractional Step Method for the Accurate Approximation of Detonation Waves
SIAM Journal on Scientific Computing
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Applied Numerical Mathematics
Fourth-Order Runge---Kutta Schemes for Fluid Mechanics Applications
Journal of Scientific Computing
Numerical solution of under-resolved detonations
Journal of Computational Physics
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In this paper, we design stable and accurate numerical schemes for conservation laws with stiff source terms. A prime example and the main motivation for our study is the reactive Euler equations of gas dynamics. Furthermore, we consider widely studied scalar model equations. We device one-step IMEX (implicit-explicit) schemes for these equations that treats the convection terms explicitly and the source terms implicitly. For the non-linear scalar equation, we use a novel choice of initial data for the resulting Newton solver and obtain correct propagation speeds, even in the difficult case of rarefaction initial data. For the reactive Euler equations, we choose the numerical diffusion suitably in order to obtain correct wave speeds on under-resolved meshes. We prove that our implicit-explicit scheme converges in the scalar case and present a large number of numerical experiments to validate our scheme in both the scalar case as well as the case of reactive Euler equations. Furthermore, we discuss fundamental differences between the reactive Euler equations and the scalar model equation that must be accounted for when designing a scheme.