Front tracking for one-dimensional quasilinear hyperbolic equations with variable coefficients

Abstract

A new front tracking method is developed for the variable coefficient equation \(u_t + \;V(x,t)f(u)_x = 0\). The method is a generalization of Dafermos' method for the constant coefficient case and is well-defined also for certain discontinuous velocity fields V. We give an explicit inequality stating the stability with respect to flux function, velocity field, and initial data. The numerical method is unconditionally stable and has linear convergence. It is well suited for numerical calculations, as is demonstrated in four examples.