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

Highlights: • Poisson-equation approach is considered for wall distance on irregular triangular grids. • A superior wall distance conversion formula is demonstrated. • A hyperbolic Poisson solver provides accurate first/second derivatives and wall distance. • A hyperbolic Poisson solver converges faster in CPU time also. 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.