A hyperbolic Poisson solver for wall distance computation on irregular triangular grids

Abstract

This paper presents an accurate and efficient Poisson solver for wall distance computations on irregular triangular grids. It is discussed that a typical formula of computing the wall distance from a numerical solution of a Poisson equation is not very accurate when solution contours are curved and the accuracy can be improved by a second-order formula derived by Taubin. However, the improved formula, as it requires second derivatives of the numerical solution, presents a challenge for a second-order method leading to zeroth-order accurate second-order derivatives on irregular grids. To overcome the difficulty, we develop a hyperbolic Poisson solver with gradients included as additional unknowns. In this method, the solution gradient can be obtained with second-order accuracy by a second-order discretization method, and therefore the second-derivatives can be obtained with first-order accuracy. Moreover, the hyperbolic method eliminates the numerical stiffness arising from the discretization of the Laplacian operator and achieves the ever-growing O(1/h) speed-up in iterative convergence, where h is a typical mesh spacing. Superior accuracy and efficiency of the developed method are demonstrated for irregular triangular grids.