Absolutely Convergent Fast Sweeping WENO Methods for Eikonal Equations and Hybrid WENO3 Method with MLP Indicator for Conservation Laws

This dissertation presents two topics in weighted essentially non-oscillatory (WENO) schemes for solving partial differential equations (PDEs). The first part focuses on fast sweeping WENO methods for Eikonal equations. Fast sweeping methods are a class of efficient iterative methods developed in the literature to solve steady-state solutions of hyperbolic PDEs. In (Zhang et al. 2006; Xiong et al. 2010), high order accuracy fast sweeping schemes based on classical WENO local solvers were developed for solving static Hamilton-Jacobi equations. However, since high order classical WENO methods (e.g., fifth order and above) often suffer from difficulties in their convergence to steady-state solutions, iteration residues of high order fast sweeping schemes with these local solvers may hang at a level far above round-off errors even after many iterations. This issue makes it difficult to determine the convergence criterion for the high order fast sweeping methods and challenging to apply the methods to complex problems. Motivated by the recent work on absolutely convergent fast sweeping method for steady-state solutions of hyperbolic conservation laws in (Li et al. 2021), we develop high order fast sweeping methods with multi-resolution WENO local solvers for solving Eikonal equations, an important class of static Hamilton-Jacobi equations. Based on such kind of multi-resolution WENO local solvers with unequal-sized sub-stencils, iteration residues of the designed high order fast sweeping methods can settle down to round-off errors and achieve the absolute convergence. In order to obtain high order accuracy for problems with singular source-point, we apply the factored Eikonal approach developed in the literature and solve the resulting factored Eikonal equations by the new high order WENO fast sweeping methods. Extensive numerical experiments are performed to show the accuracy, computational efficiency, and advantages of the new high order fast sweeping schemes for solving static Hamilton-Jacobi equations. The second part of this dissertation is a study in hybrid WENO methods with deep learning techniques for hyperbolic conservation laws. WENO schemes are a popular class of numerical methods for solving hyperbolic conservation laws. Since WENO schemes are designed to deal with problems with both complicated solution structures and discontinuities/sharp gradient regions, their sophisticated nonlinear properties and high-order accuracy require more operations than many other schemes. The methodology of hybrid methods is an effective approach to decrease the computational costs and dissipation errors of WENO schemes and achieve better resolution. One of the key components for the success of hybrid WENO schemes is the application of a robust and efficient troubled-cell indicator, which detects the computational cells where the solution loses regularity. Recently, troubled-cell indicators based on artificial neural networks (ANNs) have been developed in the literature, which have the advantage of less dependence on tunable parameters and being more robust than many traditional troubled-cell indicators, and such ANN based troubled-cell indicators have been applied to hybrid finite difference WENO schemes effectively. Motivated by these works, we develop a hybrid finite volume WENO method with an ANN based troubled-cell indicator for solving hyperbolic conservation laws. While the finite difference WENO schemes are more efficient than the finite volume WENO schemes for multidimensional problems on uniform grids, the finite volume WENO schemes have the advantage such as being flexible and easy to apply on nonuniform grids. We introduce an ANN based troubled-cell indicator by constructing a multilayer perceptron (MLP) model, one of the most common ANN models. The third-order WENO scheme is focused in this part. Extensive numerical experiments for solving various scalar equations with both convex and non-convex cases, and the Euler systems of equations on uniform and nonuniform grids of one-dimensional (1D) and two-dimensional (2D) domains, are performed to show the accuracy and nonlinear stability of the proposed hybrid finite volume WENO scheme with the MLP troubled-cell indicator. Significant accuracy improvement and computational-cost saving over the original WENO scheme are observed. Numerical experiments and comparisons with the widely-used KXRCF indicator also show the good performance of the MLP troubled-cell indicator. Although the MLP troubled-cell indicator is trained on uniform grids, it performs very well on nonuniform grids obtained by randomly perturbing uniform grids.