A Second-Order Immersed Boundary Projection Method for Elliptic and Parabolic Problems

We present a second-order accurate version of the Immersed Boundary Projection Method for imposing Dirichlet boundary conditions at irregular boundaries while using a Cartesian grid and a standard discretization, applicable to elliptic and parabolic PDEs. No modi cation to the stencil is required. Rather, boundary conditions are enforced by applying a force at the boundary that is unknown a priori. This boundary force is solved by applying additional constraints to the system, i.e. the boundary conditions desired on the immersed boundary. The accuracy of the algorithm is demonstrated by applying it to several examples arising from discretizations of Laplace’s equation and the heat equation, where second-order accuracy is obtained at all grid points.