GePUP: Generic Projection and Unconstrained PPE for Fourth-Order Solutions of the Incompressible Navier–Stokes Equations with No-Slip Boundary Conditions

Abstract

A generic projection maps one vector to another such that their difference is a gradient field and the projected vector does not have to be solenoidal. Via a commutator of Laplacian and the generic projection, the projected velocity is formulated as the sole evolutionary variable with the incompressibility constraint enforced by a pressure Poisson equation so that the dissipation of velocity divergence is governed by a heat equation. Different from previous projection methods, the GePUP formulation treats the time integrator as a black box. This prominent advantage is illustrated by straightforward formations of a semi-implicit time-stepping scheme and another explicit time-stepping scheme. Apart from its stability, the GePUP schemes have an optimal efficiency in that within each time step the solution is advanced by solving a sequence of linear systems with geometric multigrid. A key component of the GePUP schemes is a fourth-order discrete projection for no-penetration domains. Results of numerical tests in two and three dimensions demonstrate that the GePUP schemes are fourth-order accurate both in time and in space. To facilitate efficiency comparison to other methods, a simple formula is introduced. Systematic arguments and timing results show that the GePUP schemes could be vastly superior over lower-order methods in terms of efficiency and accuracy. In some cases, the GePUP schemes running on the author's personal desktop would be faster than a second-order method running on the fastest supercomputer in the world! This paper contains enough details so that one can reproduce the numerical results by following the exposition.