Immersed Boundary Smooth Extension (IBSE): A high-order method for solving incompressible flows in arbitrary smooth domains

TL;DR

The paper introduces an advanced immersed boundary method that achieves high-order accuracy for incompressible flow simulations in smooth domains by smoothly extending solutions, applicable with various discretizations.

Contribution

It extends the IBSE method to enforce divergence constraints, enabling high-order convergence for Stokes and Navier-Stokes equations in arbitrary smooth domains.

Findings

Achieves up to third-order convergence for velocity

Attains second-order convergence for stress tensor elements

Works with spectral and finite-difference discretizations

Abstract

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor. In a previous work we introduced the Immersed Boundary Smooth Extension (IBSE) method, a variation of the IB method that achieves high-order accuracy for elliptic PDE 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. In this work, we extend the IBSE method to allow for the imposition of a divergence constraint, and demonstrate high-order convergence for the Stokes and incompressible Navier-Stokes equations: up to third-order pointwise convergence for the velocity field, and second-order pointwise convergence for all elements of the stress tensor. The method is flexible to the underlying discretization: we demonstrate solutions produced using both a Fourier spectral discretization and a standard second-order finite-difference discretization.