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<p>For solving hyperbolic conservation laws that arise frequently in computational physics, high order finite volume WENO (FV-WENO) schemes and discontinuous Galerkin (DG) methods are more popular because of their applicability to any monotone fluxes and easily handle complicated geometries and boundary conditions. However, when there are smaller scale structures in the flow field, the classic FV-WENO schemes will produce a significant oscillation at small scale discontinuities (or high gradients).. This phenomenon also exists in DG methods that use nonlinear WENO limiters (DG-WENO), and this will disrupt the stability of numerical methods. In this study, a simple, robust, and effective affine-invariant finite volume WENO (FV-Ai-WENO) scheme under <i>nonuniform</i> Cartesian meshes is devised, motivated by the technique in [Don et al., J. Comput. Phys., 2022, 448: 110724]. We prove and validate that for any given sensitivity parameter, the WENO operator and the affine transformation operator in the present schemes are commutable. In the presence of smaller scale discontinuities, the new operator satisfies the ENO property while the classic WENO operator does not. In addition, we investigate using FV-Ai-WENO methodology as limiters for the DG methods, to obtain a robust, high-order accuracy, high-resolution and nonoscillatory shock transition for DG methods, especially in numerical simulations of flow fields with small scale discontinuous (or high gradient) structures. One- and two-dimensional classic examples are used to verify the performance of the FV-Ai-WENO schemes and DG methods with the Ai-WENO limiter (DG-Ai-WENO) in terms of accuracy, robustness, and affine invariance.</p>

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Affine-Invariant WENO Operator on Nonuniform Cartesian Mesh with Application to Finite Volume and Discontinuous Galerkin Methods

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Xiao-Shuo Xiang,
  • Bao-Shan Wang,
  • Zhen Gao,
  • Peng Li

摘要

For solving hyperbolic conservation laws that arise frequently in computational physics, high order finite volume WENO (FV-WENO) schemes and discontinuous Galerkin (DG) methods are more popular because of their applicability to any monotone fluxes and easily handle complicated geometries and boundary conditions. However, when there are smaller scale structures in the flow field, the classic FV-WENO schemes will produce a significant oscillation at small scale discontinuities (or high gradients).. This phenomenon also exists in DG methods that use nonlinear WENO limiters (DG-WENO), and this will disrupt the stability of numerical methods. In this study, a simple, robust, and effective affine-invariant finite volume WENO (FV-Ai-WENO) scheme under nonuniform Cartesian meshes is devised, motivated by the technique in [Don et al., J. Comput. Phys., 2022, 448: 110724]. We prove and validate that for any given sensitivity parameter, the WENO operator and the affine transformation operator in the present schemes are commutable. In the presence of smaller scale discontinuities, the new operator satisfies the ENO property while the classic WENO operator does not. In addition, we investigate using FV-Ai-WENO methodology as limiters for the DG methods, to obtain a robust, high-order accuracy, high-resolution and nonoscillatory shock transition for DG methods, especially in numerical simulations of flow fields with small scale discontinuous (or high gradient) structures. One- and two-dimensional classic examples are used to verify the performance of the FV-Ai-WENO schemes and DG methods with the Ai-WENO limiter (DG-Ai-WENO) in terms of accuracy, robustness, and affine invariance.