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The Immersed Boundary Method (IBM) is a popular numerical approach to imposeboundary conditions without relying on body-fitted grids, thus reducing thecostly effort of mesh generation. To obtain enhanced accuracy, IBM can becombined with high-order methods (e.g., discontinuous Galerkin). For thiscombination 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 thecombination of IBM (based on volume penalization) and high-order methods (basedon nodal discontinuous Galerkin methods) to analyze a priori numerical errorsand obtain practical guidelines on the selection of IBM parameters. Theanalysis is performed on a linear advection-diffusion equation with Dirichletboundary conditions. Three ways to penalize the immerse boundary areconsidered, the first penalizes the solution inside the IBM region (classicapproach), whilst the second and third penalize the first and secondderivatives of the solution. We find optimal combinations of the penalizationparameters, including the first and second penalizing derivatives, resulting inminimum errors. We validate the theoretical analysis with numerical experimentsfor one- and two-dimensional advection-diffusion equations.

computers and fluids

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