Fourth-order accurate compact-difference discretization method for Helmholtz and incompressible Navier-Stokes equations

A fourth-order accurate solution method for the three-dimensional Helmholtz equations is described that is based on a compact finite-difference stencil for the Laplace operator. Similar discretization methods for the Poisson equation have been presented by various researchers for Dirichlet boundary conditions. Here, the complicated issue of imposing Neumann boundary conditions is described in detail. The method is then applied to model Helmholtz problems to verify the accuracy of the discretization method. The implementation of the solution method is also described. The Helmholtz solver is used as the basis for a fourth-order accurate solver for the incompressible Navier-Stokes equations. Numerical results obtained with this Navier-Stokes solver for the temporal evolution of a three-dimensional instability in a counter-rotating vortex pair are discussed. The time-accurate Navier-Stokes simulations show the resolving properties of the developed discretization method and the correct prediction of the initial growth rate of the three-dimensional instability in the vortex pair.