Efficient Runge-Kutta Based Local Time-Stepping Methods

Abstract

The method of lines approach to the numerical solution of transient hyperbolic partial differential equations (PDEs) allows us to write the PDE as a system of ordinary differential equations (ODEs) in time. Solving this system of ODEs explicitly requires choosing a stable time step satisfying the Courant-Friedrichs-Lewy (CFL) condition. When a uniform mesh is used, the global CFL number is used to choose the time step and the system is advanced uniformly in time. The need for local time-stepping, i.e., advancing elements by their maximum locally defined time step, occurs when the elements in the mesh differ greatly in size. When global time-stepping is used, the global CFL number and the globally defined time step are defined by the smallest element in the mesh. This leads to inefficiencies as a few small elements impose a restrictive time step on the entire mesh. Local time-stepping mitigates these inefficiencies by advancing elements by their locally defined time step and, hence, reduces the number of function evaluations.

In this thesis, we present two local time-stepping algorithms based on a third order Runge-Kutta method and the classical fourth order Runge-Kutta method. We prove these methods keep the order of accuracy of the underlying Runge-Kutta methods in the context of a system of ODEs. We then show how they can be used with the method of lines approach to the numerical solution of PDEs, specifically with the discontinuous Galerkin (DG) spatial discretization. Numerical simulations show we obtain the theoretical $p+1$ rate of convergence of the DG method in both the $L^2$ and maximum norms. We provide evidence that these algorithms are stable through a number of linear and nonlinear examples.