High Order Strong Stability Preserving Time Discretizations for the Time Evolution of Hyperbolic Partial Differential Equations

Abstract : The objective of this project is to develop and analyze stable time discretizations suitable for the simulation of hyperbolic time-dependent partial differential equations. Implicit and explicit multi-step multi-stage time discretizations with optimal time-step restrictions have been developed, as well as implicit Runge--Kutta methods with downwinding and unconditional stability. A testing suite has been written to test some of these methods with a variety of spatial discretizations. New directions have been explored for implicit-explicit methods, which are useful for problems with convection and diffusion. Finally, novel provably stable multi-step time discretizations for use with Fourier pseudo-spectral spatial approximations of the three dimensional viscous Burgers equations and Navier-Stokes equations have been developed.