M.Sc. Thesis

Constructing High-Order Runge-Kutta Methods with Embedded Strong-Stability-Preserving Pairs

Keywords: Runge-Kutta methods, embedded Runge-Kutta methods, strong-stability-preserving, hyperbolic conservation laws, ENO, WENO, method of lines.


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).

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. Strong-stability-preserving (SSP) Runge-Kutta schemes are particularly well suited for use near non-smooth or discontinuous behavior such as shocks because they have a nonlinear stability property that helps them prevent spurious oscillations (such as the Gibb's phenomenon) and other non-physical behaviour. Unfortunately, SSP schemes have limitations that make them expensive or inappropriate in smooth regions of the solution where a high order of accuracy is desired. In these regions, schemes based on “classical” linear stability analysis are likely a better choice. This motivates the use of high-order Runge-Kutta schemes with embedded SSP pairs, where the higher-order scheme, based on linear stability analysis, would be used to evolve smooth regions of the solution. The lower-order SSP scheme would be used near shocks or other discontinuities to help prevent spurious oscillations. This thesis explores the construction of these new methods.

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.


Source Codes

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Butcher Tableaux

These are MATLAB codes that contain the methods from the thesis: