用于勒让德 - 高斯 - 洛巴托间断伽辽金谱元法的具有自适应网格细化的不变域保持限制
Invariant-domain-preserving limiting with Adaptive Mesh Refinement for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral Element Methods
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中文总结 AI 辅助
针对LGL - DGSEM在笛卡尔网格上的非协调界面，提出不变域保持处理方法，扩展凸限制等框架到含悬挂节点网格，推导低阶界面通量，引入稀疏化策略，结合高阶离散化、限制和自适应网格细化，通过数值验证和模拟展示方案特性。
中文摘要 AI 辅助
我们提出了一种针对勒让德 - 高斯 - 洛巴托间断伽辽金谱元法（LGL - DGSEM）在笛卡尔网格上具有自适应网格细化（AMR）的非协调界面的不变域保持（IDP）处理方法。该方法将最近为DGSEM开发的凸限制和图粘性框架扩展到包含悬挂节点的网格。从保守的砂浆公式出发，推导满足不变域保持离散化要求的低阶界面通量。引入基于LGL子单元特征函数的稀疏化策略以避免与完全连接的砂浆耦合相关的过度扩散。所得砂浆通量保持保守，在匹配界面上简化为标准协调公式，并自然适用于基于图粘性的用于凸限制的低阶格式。该构造为在非线性双曲守恒律的统一框架内结合高阶DGSEM离散化、不变域保持限制和自适应网格细化提供了缺失的要素。我们对所提方案的性质进行了数值验证，并进行了需要正性限制和激波捕捉的具有挑战性的模拟。
英文摘要
We present an invariant-domain-preserving (IDP) treatment of nonconforming interfaces for Legendre--Gauss--Lobatto Discontinuous Galerkin Spectral Element Methods (LGL-DGSEM) with adaptive mesh refinement (AMR) on Cartesian meshes. The proposed methodology extends recently developed convex limiting and graph-viscosity frameworks for DGSEM to meshes containing hanging nodes. Starting from a conservative mortar formulation, we derive low-order interface fluxes that satisfy the requirements of invariant-domain-preserving discretizations. To avoid the excessive diffusion associated with fully connected mortar couplings, a sparsification strategy based on LGL subcell characteristic functions is introduced, yielding compact interface stencils. The resulting mortar fluxes remain conservative, reduce to the standard conforming formulation on matching interfaces, and naturally fit into graph-viscosity-based low-order schemes used for convex limiting. The proposed construction provides the missing ingredient required to combine high-order DGSEM discretizations, invariant-domain-preserving limiting, and adaptive mesh refinement within a unified framework for nonlinear hyperbolic conservation laws. We provide numerical verifications of the properties of the proposed scheme and run challenging simulations that require positivity limiting and shock-capturing.
