How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier–Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the Laplacian and Leray projection operators. Here we make use of this formula to (a) improve the accuracy of computing pressure in two kinds of existing time-discrete projection methods implicit in viscosity only, and (b) devise new higher-order accurate time-discrete projection methods that extend a slip-correction idea behind the well-known finite-difference scheme of Kim and Moin. We test these schemes for stability and accuracy using various combinations of C0 finite elements. For all three kinds of time discretization, one can obtain third-order accuracy for both pressure and velocity without a time-step stability restriction of diffusive type. Furthermore, two kinds of projection methods are found stable using piecewise-linear elements for both velocity and pressure.

In this article the authors present a new solver for the Poisson problem on spherical shells. The method is derived by application of the Fast Fourier Transform (FFT) in all three spherical variables. Through a particular change of variables valid on spherical shells, this new approach avoids a variable coefficient on the radial differential operator, resulting in a constant-coefficient formulation that can be fast-diagonalized via FFT. The authors describe the Fourier expansion in the angular variables and the fast diagonalization of the radial derivative using the FFT, including second-, fourth- and sixth-order finite difference approximations. An algorithmic description of the sixth-order solver is described, and numerical tests that demonstrate the theoretical rate of convergence are presented.

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and exces-sively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uni-formly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyper-bolic system adequate for the use of modern shock capturing methods, and the other a linear hyperbolic system which contains the stiff acoustic dynam-ics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system, and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompress-ible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.

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