A modified equation analysis for immersed boundary methods based on volume penalization: applications to linear advection-diffusion and high-order discontinuous Galerkin schemes

TL;DR

This paper introduces a modified equation analysis for immersed boundary methods combined with high-order discontinuous Galerkin schemes, providing guidelines for parameter selection to minimize numerical errors in advection-diffusion problems.

Contribution

It applies a novel modified equation analysis to volume penalization-based IBM with high-order methods, offering practical error estimates and optimal parameter choices.

Findings

Optimal penalization parameters reduce numerical errors.

The analysis guides effective parameter selection for IBM.

Numerical experiments validate theoretical error predictions.

Abstract

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.