Characteristics-based high-order methods with WENO reconstructions for linear and nonlinear dynamics
Date
2022
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of Delaware
Abstract
The characteristics-based methods such as arbitrary-Lagrangian-Eulerian (ALE), semi-Lagrangian (SL), and Eulerian-Lagrangian (EL) are widely used in solving hyperbolic equations and convection-dominated problems in many scientific fields. The high-order characteristics-based methods are known for their high efficiency and high resolution. However, these methods have significant obstacles when applied to nonlinear dynamics due to the nonlinear evolution and coupling of characteristics. Designing a method that can solve the nonlinear equations accurately and capture the sharp discontinuities without any physical oscillations is highly nontrivial. Under such motivation, this dissertation investigates high-order characteristics-based numerical methods with weighted essentially non-oscillatory (WENO) and essentially non-oscillatory (ENO) reconstruction techniques for solving linear and nonlinear dynamics, for nonlinear kinetic and incompressible flow models, as well as for nonlinear compressible flows such as those that involve shock waves. ☐ The first contribution of this dissertation is that we develop and implement a new SL finite difference (FD) method that incorporates the adaptive order WENO reconstructions (WENO-AO). This new SL FD WENO-AO scheme can be viewed as an improvement of the previous SL FD WENO schemes first introduced by Qiu and Shu, in whichthe linear weights are nonexistent for variable coefficient problems. The new scheme is designed to fix this gap and keep all the excellent properties.We apply the new scheme to linear advection problems, the Vlasov-Poisson system, and the incompressible Euler equations. The SL FD WENO-AO scheme improves efficiency and accuracy compared to the previous SL FD WENO schemes. ☐ The second contribution of this dissertation is that we propose a new EL Runge-Kutta (RK) finite volume (FV) method for solving convection-dominated problems. The proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined by the Rankine-Hugoniot condition. The scheme can be viewed as a generalization of the standard RK FV scheme for which the space-time region is partitioned by approximate characteristics with zero velocity. The high-order spatial accuracy is achieved using the WENO-AO, and explicit RK methods obtain the high-order temporal accuracy. In addition, the new scheme extends to higher dimensions via dimensional splitting. Numerical experiments demonstrate the scheme's robustness, high-order accuracy, and ability to handle extra-large time steps. ☐ The final contribution of this dissertation is the most critical component in handling shocks. We construct a novel forward EL FV method for solving the nonlinear hyperbolic equations with shocks. This new method adopts the space-time regions partitioned by the forward partition lines determined by the Rankine-Hugoniot condition to relax the time-stepping size constraint. Also, to overcome the characteristics intersecting situations, we embed a cell-merging mechanism into the proposed forward EL FV method. The proposed first-order scheme is theoretically proved to be total-variation-diminishing and maximum-principle-preserving under a large time-stepping size. Numerical tests achieve excellent performances of the proposed method.
Description
Keywords
Conservation law, Eulerian-Lagrangian, Finite volume, Numerical analysis, Weighted essentially non-oscillatory