The purpose of this paper is to numerically solve Eikonal and Hamilton‐Jacobi equations using the finite element method; to use both explicit Taylor Galerkin (TG) and implicit methods to obtain shortest wall distances; to demonstrate the implemented methods on some realistic problems; and to use iterative generalized minimal residual method (GMRES) method in the solution of the equations.
The finite element method along with both the explicit and implicit time discretisations is employed. Two different forms of governing equations are also employed in the solution. The Eikonal equation in its original form is used in the explicit Taylor Galerkin discretisation to save computational time. For implicit method, however, the convection‐diffusion form in its conservation form is used to maintain spatial stability.
The finite element solution obtained is both accurate and smooth. As expected the implicit method is much faster than the explicit method. Though the proposed finite element solution procedures in serial is slower than the standard search procedure, they are suitable to be used in a parallel environment.
The finite element procedure for Eikonal and Hamilton‐Jacobi equations are attempted for the first time. Though the finite volume and finite difference‐based computational fluid dynamics (CFD) solvers have started employing differential equations for wall distance calculations, it is not common for finite element solvers to use such wall distance calculations. The results presented here clearly show that the proposed methods are suitable for unstructured meshes and finite element solvers.
