Khalida Parveen | Applied Math, University of Waterloo
Explicit Runge-Kutta time-stepping with the discontinuous Galerkin method
In this thesis, the discontinuous Galerkin method is used to solve the hyperbolic equations.
The DG method discretizes a system into 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 by 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.