High-order multiderivative methods for ordinary and partial differential equations

Abstract

This thesis concerns the design, analysis and application of time discretization applied to differential equations. We work with multiderivative techniques for the time derivatives. We use high-order temporal derivatives to obtain higher-order methods rather than multiple stages or multiple steps. First, we shed light on an issue that arises with implicit-explicit multiderivative schemes. We develop and demonstrate that a novel splitting fully preserves the scheme’s explicitness while not degrading the order of convergence. Moreover, we enforce the functional preservation in ordinary differential equations using a relaxation technique. This technique can be implemented with relatively low computational cost. By doing this, we considerably reduce the error growth for the numerical method, especially when the simulation time was long. Additionally, we use an implicit two-derivative temporal discretization scheme for the Cahn-Hilliard equation. We show analytically and numerically that the multiderivative techniques preserve the energy stability of the system. This is the first time an energy-stable multiderivative approach is established for the Cahn-Hilliard equation. Finally, adding stabilization terms, we prove that an implicit two-derivative approach is energy stable.