Explicit Runge-Kutta time-stepping with the discontinuous Galerkin method

Abstract

In this thesis, the discontinuous Galerkin method is used to solve the hyperbolic equations. The DG method discretizes a system into a semi-discrete system and a system of ODEs is obtained. To solve this system of ODEs efficiently, numerous time-stepping techniques can be used. The most popular choice is Runge-Kutta methods. Classical Runge-Kutta methods need a lot of space in the computer memory to store the required information. The 2N-storage time-steppers store the values in two registers, where N is the dimension of the system. The 2N-storage schemes have more stages than classical RK schemes but are more efficient than classical RK schemes.

Several 2N-storage time-stepping techniques have been used reported in the literature. The linear stability condition is found using the eigenvalue analysis of DG method and spectrum of DG method has been scaled to fit inside the absolute stability regions of 2N-storage schemes. The one-dimensional advection equation has been solved using RK-DG pairings. It is shown that these high-order 2N-storage RK schemes are a good choice for use with the DG method to improve efficiency and accuracy over classical RK schemes.