Journal of Computational Physics
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.
Complex geometry, Embedded boundary, Fourier spectral method, High-order, Immersed boundary, Incompressible Navier–Stokes
Stein, David B.; Guy, Robert D.; and Thomases, Becca, "Immersed Boundary Smooth Extension (IBSE): A High-Order Method for Solving Incompressible Flows in Arbitrary Smooth Domains" (2017). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.