Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions∗
1 University of
Milano-Bicocca, Via Cozzi
2 University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany.
Revised: 1 July 2015
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.
Mathematics Subject Classification: 65M12 / 35M30
Key words: Numerical analysis / mixed systems of partial differential equations / coupled equations / Lax−Friedrichs method / finite difference schemes / nonlocal conservation laws
The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabiliteà le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart.
© EDP Sciences, SMAI 2016