Abstract

In this work we explore a velocity correction method that introduces the splitting at the discrete level. In order to do so, the algebraic continuity equation is transformed into a discrete pressure Poisson equation and a velocity extrapolation is used. In Badia et al. (IJNMF, 2008, p. 351), where the method was introduced, the discrete Laplacian that appears in the pressure Poisson equation is approximated by a continuous one using an extrapolation for the pressure. In this work we explore the possibility of actually solving the discrete Laplacian. This introduces significant differences because the pressure extrapolation is avoided and only a velocity extrapolation is needed. Our numerical results indicate that it is the second‐order pressure extrapolation which makes third‐order methods unstable. Instead, second‐order velocity extrapolations do not lead to instabilities. Avoiding the pressure extrapolation allows to obtain stable solutions in problems that become unstable when the Laplacian is approximated. A comparison with a pressure correction scheme is also presented to verify the well‐known fact that the use of a second order pressure extrapolation leads to instabilities. Therefore we conclude that it is the combination of a velocity correction scheme with a discrete Laplacian that allows to obtain a stable third‐order scheme by avoiding the pressure extrapolation.

Abstract

For residual-based stabilization methods such as streamline-upwind Petrov–Galerkin (SUPG) and finite calculus (FIC), the higher-order derivatives of the residual that appear in the stabilization term vanish when simplicial elements are used. The sub-grid scale method using orthogonal sub-scales (OSS) attempts to recover the lost consistency by using a fine-scale projected residual in the stabilization term. The FIC method may also be cast into an OSS form with very little manipulation using an auxiliary convective projection equation. This paper discusses the gain/loss by recovering the consistency of the discrete residual in the stabilization terms via the form that includes the convective projection (as in the OSS method). We present the von Neumann analysis of the FIC method with recovered consistency (FIC RC) for the 1D convection–diffusion problem and we compare it with the standard Bubnov–Galerkin linear finite element method and FIC/SUPG methods. The transient analysis is done by examining the discrete dispersion relation of the stabilization methods. The spectral results for the semi-discrete and fully discrete problem are presented with time integration done by the trapezoidal and second-order backward differencing formula schemes. The effect of lumping the effective mass matrix T is considered relative to using a consistent form. The effect of refinement in space and time is also discussed. Finally, an optimal expression for the stabilization parameter for the FIC RC method on a uniform grid and for the steady state is given and its performance in the transient mode is discussed.