Published in Int. Journal for Numerical Methods in Fluids, Vol. 74 (2), pp. 872-897, 2014
doi: 10.1002/fld.3877
Abstract
This paper aims at the development of a new stabilization formulation based on the Finite Calculus (FIC) scheme for solving the Euler equations using the Galerkin finite element method (FEM) on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space-time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the spacetime domain. An explicit fourth order Runge-Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high gradient zones.
