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.

本学位论文围绕求解偏微分方程（PDEs）的加权本质无振荡（WENO）格式，展开了两个主题的研究。第一部分聚焦于程函方程的快速扫描WENO方法。快速扫描方法是一类已在文献中提出的高效迭代方法，用于求解双曲型偏微分方程的稳态解。在Zhang等（2006）、Xiong等（2010）的研究中，基于经典WENO局部求解器的高精度快速扫描格式被提出，用于求解静态哈密顿-雅可比方程。然而，由于高阶经典WENO方法（如五阶及以上）在收敛至稳态解时往往存在困难，采用此类局部求解器的高阶快速扫描格式的迭代残差即使经过多轮迭代，仍可能停滞在远高于舍入误差的水平。该问题使得高阶快速扫描方法的收敛准则难以确定，且难以将其应用于复杂问题。受Li等（2021）中针对双曲守恒律稳态解的绝对收敛快速扫描方法的最新研究启发，我们针对一类重要的静态哈密顿-雅可比方程——程函方程，提出了采用多分辨率WENO局部求解器的高阶快速扫描方法。基于这类具有不等长子模板的多分辨率WENO局部求解器，所设计的高阶快速扫描方法的迭代残差可收敛至舍入误差水平，实现绝对收敛。针对含奇异源点的问题，为保证其计算精度，我们采用文献中提出的因式分解程函方法，并通过新提出的高阶WENO快速扫描方法求解得到的因式分解程函方程。我们开展了大量数值实验，以验证所提出的高阶快速扫描格式在求解静态哈密顿-雅可比方程时的精度、计算效率与优势。
本学位论文的第二部分，是针对双曲守恒律的融合深度学习技术的混合WENO方法研究。WENO格式是一类广泛使用的双曲守恒律数值求解方法。由于WENO格式被设计用于处理兼具复杂解结构与间断/陡梯度区域的问题，其复杂的非线性特性与高精度特性所需的计算量远高于其他诸多格式。混合方法的思路是一种有效降低WENO格式计算成本与耗散误差、提升求解分辨率的途径。混合WENO方法成功的关键要素之一，是采用鲁棒且高效的问题单元指示器，该指示器用于检测解正则性丢失的计算单元。近年来，基于人工神经网络（ANNs）的问题单元指示器被提出，相较于诸多传统指示器，此类指示器具有可调参数更少、鲁棒性更强的优势，且已被有效应用于混合有限差分WENO格式中。受此类研究启发，我们针对双曲守恒律，提出了一种融合基于人工神经网络的问题单元指示器的混合有限体积WENO方法。尽管在结构化均匀网格的多维问题中，有限差分WENO格式相较于有限体积WENO格式计算效率更高，但有限体积WENO格式具有更灵活、更易于在非结构化网格上应用的优势。我们通过构建多层感知机（MLP）模型——一类最常见的人工神经网络模型——来实现基于人工神经网络的问题单元指示器。本部分的研究以三阶WENO格式为对象。我们开展了大量数值实验，涵盖一维与二维（2D）结构化/非结构化网格下的凸、非凸标量方程，以及欧拉方程组的求解场景，以验证所提出的融合MLP问题单元指示器的混合有限体积WENO格式的精度与非线性稳定性。实验结果表明，相较于原始WENO格式，该方法在求解精度与计算成本节约上均有显著提升。通过与广泛使用的KXRCF指示器进行数值实验与对比，也验证了MLP问题单元指示器的优异性能。尽管MLP问题单元指示器仅在均匀网格上训练，但其在通过均匀网格随机扰动得到的非均匀网格上仍表现出色。