A Chebyshev-Galerkin method for inertial waves

Abstract

We consider the incompressible viscous Navier-Stokes equations for a rotating fluid. In a tilted domain inertial waves converge to a wave-attractor and influence the mean flow. This motivates the numerical simulation of the solution to the Navier-Stokes equations with a Chebyshev-Galerkin method. A Stokes time-marching scheme involves two second-order partial differential equations and guarantees continuity.

We develop a Chebyshev-Galerkin method to find a weak solution to a set of separable second-order partial differential equations in a three-dimensional rectangular domain with homogeneous boundary conditions. We consider a spectral method with Chebyshev polynomials as basis functions. Weak solutions of a second-order partial differential equation are obtained by solving a linear system of inner-product matrices, which can be solved in terms of expansion coefficients. Boundary conditions can be satisfied with a superposition of Chebyshev polynomials. This approach is extended to higher dimensions in rectangular domains for separable operators.

Chebyshev polynomials are orthogonal with respect to a weighted inner-product. Thus, elegant expressions exist for the mass, first-derivative and stiffness matrices. We show exponential convergence of accuracy with grid-resolution and design fast schemes with linear complexity for the multiplication of the inner-product matrices. We construct diagonal preconditioners for the Laplace-operator in one, two and three dimensions. The preconditioned Poisson-system has a condition number that increases sublinearly with grid-size. This is shown analytically in one dimension and numerically in up to three dimensions.