Simple, accurate treatment of curved boundaries with dirichlet and neumann conditions

A simple and accurate numerical treatment of curved boundaries for the solution of elliptic or elliptic-parabolic partial differential equations typical of transport problems is described. The study is motivated by the importance of curved boundaries in applied and fundamental problems, especially those with Neumann conditions. Existing methods, including those employing unstructured or body-fitted meshes, are either clumsy, of low-order accuracy, or burdened with high computational overhead. The method employs elementary finite-difference meshes and places nodes on the curved boundaries to permit precise satisfaction of the boundary conditions. A key element of the method is the use of skew-tolerant algorithms to alleviate possible computational difficulties when boundary nodes are too close to interior nodes. Second-order equations and implementations are presented, easily extendable to third or higher orders. The higher-order space-time formulations and the implicit time integration's freedom from stability constraints render the approach potentially quite economical compared to conventional first-order approaches. The ability to satisfy Neumann boundary conditions accurately and economically can be a great asset in many problems, including free-boundary problems involving phase change and/or surface tension, as well as in pressure computations in Navier-Stokes solvers.