Implicit-Explicit Multivalue Methods for semi-discretized PDEs

Many practical problems in science and engineering are modeled by PDEs, whose space-discretization gives raise to large systems of ordinary differential equations (ODEs), characterized by a stiff part and a non-stiff one. A signifcant example is given by the discretization of advection-diffusion equations or advection-reaction problems with a stiff reaction. Here we propose, for the numerical solution of such systems, implicit-explicit multivalue methods which treat the non-stff part by an explicit multivalue method and the stiff part by a diagonally implicit multivalue method. We analyze the convergence and stability of these methods when implicit and explicit parts interact with each other. We look for methods with large absolute stability region, assuming that the implicit part of the method is A- or L-stable. Moreover we are interested in IMEX methods which are able to reproduce the qualitative behavior of the solution. We furnish examples of some IMEX methods, with optimal stability properties. Finally, we compare our methods with other existing ones, on some significant test examples.