High order finite volume schemes for balance laws with stiff relaxation

Abstract The paper deals with the construction and analysis of efficient high order finite volume shock capturing schemes for the numerical solution of hyperbolic systems with stiff relaxation. In standard high order finite volume schemes it is difficult to treat the average of the source implicitly, since the computation of such average couples neighboring cells, making implicit schemes extremely expensive. The main novelty of the paper is that the average of the source is split into the sum of the source evaluated at the cell average plus a correction term. The first term is treated implicitly, while the small correction is treated explicitly, using IMEX-Runge–Kutta methods, thus resulting in a very effective semi-implicit scheme. This approach allows the construction of effective high order schemes in space and time. An asymptotic analysis is performed for small values of the relaxation parameter, giving an indication on the structure of the IMEX schemes that have to be adopted for time discretization. Several numerical tests confirm the accuracy and efficiency of the approach.