Implicit-explicit and explicit projection schemes for the unsteady incompressible Navier–Stokes equations using a high-order dG method

Abstract

Abstract A modified version of the projection scheme [19] is proposed, which does not show a lower limit for the time step in contrast to the limits of stability observed numerically for some projection type schemes. An advantage of the proposed scheme is that the right-hand side of the Poisson equation for the pressure is independent of the time step. An explicit version of the current scheme is also provided besides the implicit-explicit one. For the implicit-explicit version, we retain divergence of the viscous terms on the right-hand side of the Poisson equation in order to achieve a higher accuracy for low Reynolds number flows. In this way, we also ensure that the Poisson equation with Neumann boundary condition is consistent on the discrete level, where we discretize the boundary condition as well. The spatial discretization is performed for equal and mixed orders of the velocity and pressure using the dG method. Long-term stability and optimal temporal and spatial convergence rates are obtained for the unsteady Taylor vortex and periodic channel flows. An excellent agreement with the similarity solution is achieved for the steady plane stagnation point flow. Finally, the current projection scheme is compared with the original one for a more complex coupled electro-fluid-dynamics problem. For the same numerical settings, the current scheme appears to be more accurate in predicting the expected physical behavior and it is long-term stable while the original scheme fails.