STRUCTURE-OPTIMIZED AND ORDER-PRESERVING METHODS FOR ADVECTION AND TIME STEPPING

Abstract

This dissertation addresses some of the existing fundamental shortcomings in computational methods for differential equations by developing novel high-order accurate numerical schemes that can apply to multi-physics or multi-scale problems. Motivated by the need for numerical methods with long-time accurate solutions in complex advection-dominated problems, we first address whether there are numerical schemes for constant-coefficient advection problems that can yield convergent solutions for an infinite time horizon. After establishing a new notion of convergence in an infinite time limit of numerical methods, we first show that linear methods on a fixed-grid cannot meet this convergence criterion. Also, numerical studies demonstrate that some widely used traditional nonlinear methods fail to produce accurate long-time solutions. Then we present a new numerical methodology based on a nonlinear jet scheme framework. We show that these methods satisfy the new convergence criterion, establishing that such numerical methods exist that converge on an infinite time horizon and demonstrate the long-time accuracy gains incurred by this property. Secondly, we develop finite volume methods with optimal limiter functions based on the class of WENO type methods. Limiter functions are applied to finite volume methods to avoid oscillations near discontinuities and sharp transitions for hyperbolic equations. Many limiter functions have been introduced in the literature.The general approach is first to devise a limiter function and then demonstrate that it performs well on some test problems. Here, we solve the inverse problem instead: given a portfolio of representative test cases, and a cost functional, determine the optimal limiter function that is then designed to do well. Lastly, we address the question of whether diagonally implicit Runge-Kutta (DIRK) schemes with weak stage order (WSO) four or higher and order at least four exist. The concept of WSO, which manifests in algebraic conditions on the Runge-Kutta Butcher tableau, can relax the stage order conditions so that it becomes compatible with the DIRK structure. Thus, DIRK schemes with high WSO can rectify order reduction. DIRK schemes up to order four with WSO up to three have already been constructed, based on a simplified theory. In this part of the dissertation, we develop a general theory of WSO and a methodology that can yield stiffly accurate, A-stable DIRK schemes up to order five with WSO up to five.