Multidimensional Fem-tvd Paradigm for Convection-dominated Flows

An algebraic approach to the design of multidimensional high-resolution schemes is introduced and elucidated in the finite element context. A centered space discretization of unstable convective terms is rendered local extremum diminishing by a conservative elimination of negative off-diagonal coefficients from the discrete transport operator. This modification leads to an upwind-biased low-order scheme which is nonoscillatory but overly diffusive. In order to reduce the incurred error, a limited amount of compensating antidiffusion is added in regions where the solution is sufficiently smooth. A node-oriented flux limiter of TVD type is designed so as to control the ratio of upstream and downstream edge contibutions to each node. Nonlinear algebraic systems are solved by an iterative defect correction scheme preconditioned by the low-order evolution operator which enjoys the M-matrix property. The diffusive and antidiffusive terms are represented as a sum of antisymmetric internodal fluxes which are constructed edge-by-edge and inserted into the global vectors. The new methodology is applied to the equations of the k−ε turbulence model and relevant implementation aspects are addressed. Numerical results are presented for the three-dimensional incompressible flow over a backward facing step.