Fourth order accurate numerical solution of the sine-Gordon equation : using the summation-by-parts simultaneous approximation term method

Abstract: This project deals with creating a numerical solver of the sine-Gordon equation using the summation- by- parts and simultaneous approximation term method in combination with a finite difference time- stepping method as well as a Runge-Kutta time-stepping method. All implementations were done with fourth order accuracy and the theoretical work involved in deriving such a finite difference time-stepping method for the sine-Gordon equation is presented. Both the finite difference and the Runge-Kutta time- stepping methods conserved the energy of the solutions. The only significant difference between the two time-stepping methods was that the finite difference method executed significantly faster than the Runge-Kutta method. However, the Runge- Kutta method is easier to implement and may therefore be preferable when execution time is non-vital.