Numerical Methods for Mixed Type PDEs

Many PDE systems contain hyperbolic, parabolic, and elliptic modes. For example, in the classical hydrodynamic model for semiconductor devices, the hyperbolic compressible fluid dynamics of electrons is coupled to parabolic heat conduction and/or viscosity and to the elliptic Poisson equation for the electric potential. The most general approach is to use fractional timesteps: A timestep can be split into a hyperbolic transport partial timestep, followed by a parabolic heat conduction/viscosity partial timestep, with an elliptic electric potential update at the end. Detailed examples of timestep splittings will be given for the drift-diffusion model (parabolic plus elliptic), the incompressible Navier-Stokes equations (incompletely parabolic), and the classical hydrodynamic model (hyperbolic plus parabolic plus elliptic).