论文标题
基于音量惩罚的沉浸边界方法的修改方程分析：线性对流扩散和高阶不连续的Galerkin方案的应用
A modified equation analysis for immersed boundary methods based on volume penalization: applications to linear advection-diffusion and high-order discontinuous Galerkin schemes
论文作者
论文摘要
沉浸式边界方法（IBM）是一种流行的数值方法，可以强加边界条件而不依赖人体构成的网格，从而减少了网格生成的昂贵努力。为了获得增强的精度，可以将IBM与高阶方法（例如不连续的Galerkin）结合使用。为了使这种组合有效，对数值错误的分析至关重要。在这项工作中，我们首次将修改的方程分析应用于IBM（基于体积惩罚）和高阶方法（基于淋巴结不连续的Galerkin方法）的组合，以分析先验的数值错误，并获得有关IBM参数选择的实用准则。该分析是对具有差异元素边界条件的线性对流扩散方程进行的。考虑了惩罚浸入边界的三种方法，第一个惩罚了IBM区域内的解决方案（经典方法），而第二和第三则惩罚了解决方案的第一个和第二个衍生物。我们发现惩罚参数的最佳组合，包括第一个和第二个惩罚衍生物，从而导致最小错误。我们通过数值实验对一维对流扩散方程进行验证理论分析。
The Immersed Boundary Method (IBM) is a popular numerical approach to impose boundary conditions without relying on body-fitted grids, thus reducing the costly effort of mesh generation. To obtain enhanced accuracy, IBM can be combined with high-order methods (e.g., discontinuous Galerkin). For this combination to be effective, an analysis of the numerical errors is essential. In this work, we apply, for the first time, a modified equation analysis to the combination of IBM (based on volume penalization) and high-order methods (based on nodal discontinuous Galerkin methods) to analyze a priori numerical errors and obtain practical guidelines on the selection of IBM parameters. The analysis is performed on a linear advection-diffusion equation with Dirichlet boundary conditions. Three ways to penalize the immerse boundary are considered, the first penalizes the solution inside the IBM region (classic approach), whilst the second and third penalize the first and second derivatives of the solution. We find optimal combinations of the penalization parameters, including the first and second penalizing derivatives, resulting in minimum errors. We validate the theoretical analysis with numerical experiments for one- and two-dimensional advection-diffusion equations.
