A hybrid finite difference--finite volume approach to solve first-order hyperbolic conservation laws with superior accuracy

Abstract

A hybrid finite difference--finite volume (FD-FV) approach for discretization in space is proposed to solve first-order hyperbolic conservation laws. Unlike any conventional finite difference method (FDM) or finite volume method (FVM), this approach uses both cell-averaged values and nodal values as degrees of freedom (DOF). Consequently it is inherently conservative like FVM and easy to extend to high-order accuracy in space like FDM. The proposed FD-FV approach works for arbitrary flux functions, whether convex or non-convex; and it does not require any exact or approximate Riemann solver hence it is also computationally economical. Method of lines is adopted for time integration in present work; in particular, explicit Runge-Kutta methods are employed. It is theoretically proven and numerically confirmed that in general, the proposed FD-FV methods possess superior accuracy than conventional FDM or FVM. Linear stability is studied for general FD-FV schemes -- both space-accurate and time-stable FD-FV schemes of up to fifth-order accuracy in both space and time are presented. Numerical examples show that as long as the solutions are smooth, the proposed FD-FV methods are more efficient than conventional FVM of the same order, at least when explicit time-integrators are applied.