Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

TL;DR

The paper introduces the IBSE method, a high-order numerical scheme that extends solutions smoothly to enable Fourier spectral methods for PDEs on complex domains, improving accuracy over traditional immersed boundary approaches.

Contribution

It presents a novel high-order extension technique for PDEs on arbitrary smooth domains, compatible with spectral methods, requiring minimal geometric boundary information.

Findings

Achieves fourth-order convergence for Dirichlet problems

Achieves third-order convergence for Neumann problems

Demonstrates effectiveness on various PDEs like Poisson and heat equations

Abstract

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems.