Upwinding with deferred correction (UPDC): an effective implementation of higher-order convection schemes for implicit finite volume methods

Abstract

A conceptually simple and computationally inexpensive implementation of higher-order convection interpolation schemes for stable and faster convergence of the implicit finite volume method (FVM) is evaluated for multi-dimensional viscoelastic flow calculations as well as convection-dominated Newtonian flow calculations. Based on the deferred-correction method, this effective formulation consistently satisfies the four rules that guarantee bounded numerical solutions. In addition, artificial numerical diffusion, commonly found in numerical solutions with the first-order upwinding convection scheme, can be effectively controlled by consistently using a third-order quadratic upstream interpolation over the entire computational domain. Extensive tests performed for the steady-state wall-driven square enclosure flow of a Newtonian fluid at Reynolds number ( Re) up to 5000 demonstrate that the formulation is stable, accurate, and converges well. The accuracy of the formulation for multi-dimensional viscoelastic flow calculations is evaluated by solving the Oldroyd-B equations for the problem of a fully two-dimensional steady frictionless plane flow, and comparing the numerical results with the exact solution available. With the first-order upwinding scheme, the results show that, for hyperbolic constitutive models, regardless of the grid size, there always exists artificial diffusion whenever the flow streamlines are not closely aligned with the grid lines. With the new implementation, artificial numerical diffusion is effectively controlled and the results demonstrate third-order accuracy.