| Runge-Kutta methods are one of the fundamental techniques in scientific computing. They are used to compute numerical solutions in a step-by-step fashion for ordinary differential equations (ODEs) and also, via the method of lines, for partial differential equations (PDEs).;Following a review of Runge-Kutta methods, strong-stability, and other related concepts, the proprietary BARON optimization software is introduced as a powerful tool for deriving optimal SSP schemes. Various Runge-Kutta methods with embedded SSP pairs are then constructed using a combination of BARON optimization and analytical techniques. (Abstract shortened by UMI.);By sharing information, embedded Runge-Kutta methods execute two Runge-Kutta schemes simultaneously while incurring minimal additional cost. Traditionally this is done for the purpose of actively selecting step-sizes for error control. However, in this thesis, we suggest another possible use where the two schemes would be used in different regions of the spatial domain based on local properties of the solution. For example, the solutions of hyperbolic conservation laws contain both smooth and non-smooth features. |
