Second-order accurate finite volume method for G -equation on polyhedral meshes

In this paper, we propose a cell-centered finite volume method to numerically solve the G -equation on polyhedral meshes in three-dimensional space, that is, a general type of the level-set equation including advective, normal, and mean curvature flow motions. The main contribution is to design a numerical algorithm for the regularized mean curvature flow equation that can be consistently combined regarding the size of the time step with previous algorithms for the advective and normal flows on polyhedral meshes. For a spatial discretization, we use a flux-balanced approximation with an orthogonal splitting of displacement vector from a center of the cell to a center of the face. For a temporal discretization, we use a nonlinear Crank–Nicolson method with a deferred correction method which gives us, firstly, second-order accuracy in space and time similarly to the algorithms for the advective and normal flow equations, and, secondly, a possibility of straightforward domain decomposition for efficient parallel computation. Numerical experiments quantitatively show that the size of time step proportional to an average size of computational cells is enough to obtain the second-order convergence in space and time for smooth solutions of the general level set equation. A qualitative comparison is presented for a nontrivial example to compare numerical results obtained with hexahedral and polyhedral meshes. Finally, an example of solving numerically the G -equation used in combustion literature is given.