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

The paper tackles numerical errors in volume-penalized immersed boundary methods coupled with high-order nodal DG (DGSEM) schemes. It introduces a modified equation analysis to identify and cancel leading spatial errors by augmenting the volume-penalization with first- and second-derivative penalties, deriving optimal parameters that minimize error inside the solid region. The analysis predicts that setting $oldsymbol{ ext{η}}_2=-1/c$ and $oldsymbol{ ext{η}}_3=1/ u$ cancels dominant dissipative and dispersive errors, with the viscous flux scheme (LDG preferred over BR1) influencing the practical accuracy in the fluid. Numerical experiments in one and two spatial dimensions validate the theoretical predictions, demonstrate reduced errors in the flow domain, and illustrate the improved performance when using LDG with the optimal penalties; the work provides a rigorous, a priori guideline for selecting VP parameters in VP-DG schemes and lays groundwork for extending the approach to nonlinear systems.

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.

Figures (10)

• Figure 1: Domain decomposition and reference interval in the DGSEM technique.
• Figure 2: Schematic illustration of the advection problem with IBM.
• Figure 3: Simulation under different penalization parameters ($r = 1/40$, $N = 3$, initial wavenumber $\overline{\omega} \Delta x/(N+1) = 0.3223$, $K = 40$): a) Global view; b) Enlarged view.
• Figure 4: Mean squared error between sharp and smooth mask function with increasing $\delta$.
• Figure 5: Error comparison for the advection equation, vertical dashed line refers to $\eta_2 = -1/c$, and horizontal dashed line refers to $\eta_2 \rightarrow \infty$. a) Error in the flow ($N=2$). b) Error in the solid, the optimal value is zero ($N=2$). c) Error in the flow ($N=3$). d) Error in the solid, the optimal value is zero ($N=3$). e) Error in the flow (larger penalization parameter, $N=3$). f) Error in the solid (larger penalization parameter, $N=3$).