Method of averaged element splittings for diffusion terms discretization in vertex-centered framework

Abstract

Method of averaged element splittings is proposed for the discretization of the diffusion terms of the Navier – Stokes equations on mixed-element unstructured meshes. It is applicable when mesh functions are defined in nodes. This method is a linear method, which has much in common with the classical P1-Galerkin method. In the case of simplicial meshes, these methods coincide, but the new method yields a 7-point approximation of the 3D Laplace operator on a Cartesian mesh. For a model elliptic problem we provide both theoretical and numerical accuracy estimates. The numerical evidence shows that accuracy of the new method is the same as of the P1-Galerkin method in 2D and possibly slightly worse in 3D. However, we only prove the order 3/2 in 2D and the first order in 3D, both in L2. The new method has an important advantage in the case of implicit time integration based on the Newton method, which implies solving linear algebraic systems with flux Jacobian. It allows to truncate flux Jacobian to a 7-point stencil for a much wider range of Reynolds and Courant numbers without loss of iterations convergence, compared to the P1-Galerkin method.