An optimal order error estimate of a linear finite element method for smooth solutions of 2D systems of conservation laws

In this paper we consider approximating smooth solutions of systems of nonlinear conservation laws by a linear finite element method with uniform mesh in two spatial dimensions, where the time discretization is carried out by a second order explicit Runge-Kutta method. An optimal error estimate O(h(2)) in L-2-norm for continuous linear finite elements is obtained under the CFL condition Delta t <= Ch(4/3), Where h and Delta t axe the spatial meshsize and the time step, respectively, and the positive constant C is independent of h and Delta t.