High-Order Numerical Methods for Time-Dependent Problems with Applications

In this dissertation, several high-order numerical methods for solving time dependent problems are studied.

In the first part, a maximum principle preserving (MPP) finite-volume (FV) weighted essentially non-oscillatory (WENO) Runge Kutta (RK) scheme is proposed for convection-dominated problems. Such problems possess the maximum principle at the theoretical level, hence it is hoped that the numerical solution preserves the maximum principle. However, normal high-order FV WENO RK scheme doesn't satisfy such property. We propose a modified high-order FV WENO scheme by adding locally-parametrized flux limiters to maintain the maximum principle. In this work, for the first time under the finite-volume framework, such flux limiters are proved to maintain the high-order accuracy of the original WENO scheme for linear advection problems without any additional time-step restriction. And for general nonlinear convection-dominated problems, the flux limiters are proved to introduce up to $O(\Delta x3+\Delta t3)$ modification to the high-order temporal integrated flux in the original WENO scheme without extra time-step constraint. The MPP property of the proposed scheme is validated by several numerical tests.

In the second part, an integral deferred correction (InDC) method with adaptive non-polynomial basis is presented to solve stiff time dependent problems whose solutions contain initial or internal layers. Several non-polynomial bases with exponential functions are proposed, in the hope that the stiff layers in the solution can be better resolved by the exponentials than by polynomials. The stability and accuracy properties of the non-polynomial InDC schemes are comparable to those of the polynomial InDC schemes. Finally, numerical test shows that the newly proposed InDC scheme outperforms the traditional polynomial-based scheme when it is applied to solve initial value problems with layers, in the sense that the former scheme takes fewer time steps than the latter one given the same error tolerance.