Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions

We prove the convergence of an explicit numerical scheme for the discretization of a coupled hyperbolic-parabolic system in two space dimensions. The hyperbolic part is solved by a Lax-Friedrichs method with dimensional splitting, while the parabolic part is approximated by an explicit finite-difference method. For both equations, the source terms are treated by operator splitting. To prove convergence of the scheme, we show strong convergence of the hyperbolic variable, while convergence of the parabolic part is obtained only weakly∗ in L∞. The proof relies on the fact that the hyperbolic flux depends on the parabolic variable through a convolution function. The paper also includes numerical examples that document the theoretically proved convergence and display the characteristic behaviour of the Lotka-Volterra equations.