EXTRAPOLATED MULTIRATE METHODS FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

Within atmospheric earth system models, accurately resolving the various physical multi-scale processes is critical. For example, the single-layered shallow water equations are widely used to model ocean waves in shallow depth, which contain both fast gravity and slow advective waves. In this dissertation we describe the development of an extrapolated multirate time-integration scheme using explicit Runge-Kutta methods for advancing the numerical solutions of hyperbolic partial differential equations with multiple processes. Our time-integration technique is based on Richardson extrapolation and will be tested using high-order accurate continuous and discontinuous Galerkin methods with an upwind-biased Rusanov flux. The benefit in developing an extrapolated multirate time-integration method is that we can solve the fast and slow processes using different time-steps with high order accuracy. Results are shown for the non-linear shallow water equations in conservation form and the non-hydrostatic, atmospheric Euler equations. We further analyze and compare the extrapolated multirate method against other single-rate and multirate time-splitting schemes for problems with multiple time scales.Lieutenant Colonel, United States ArmyApproved for public release. Distribution is unlimited