A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients

Abstract

A numerical method is presented for the variable coefficient, nonlinear hyperbolic equation u t + ∑i=1 d Vi(x, t)fi(u) x i = 0 in arbitrary space dimension for bounded velocities that are Lipschitz continuous in the x variable. The method is based on dimensional splitting and uses a recent front tracking method to solve the resulting one-dimensional non-conservative equations. The method is unconditionally stable, and it produces a subsequence that converges to the entropy solution as the discretization of time and space tends to zero. Four numerical examples are presented; numerical error mechanisms are illustrated for two linear equations, the efficiency of the method compared with a high-resolution TVD method is discussed for a nonlinear problem, and finally, applications to reservoir simulation are presented.