The "Sphered Cube": A New Method for the Solution of Partial Differential Equations in Cubical Geometry
This work offers a new numerical method for solving PDEs in cubical domains, which is particularly relevant for computational scientists working on problems in cubical geometry.
The paper introduces a volume-penalization-based gridding technique that embeds a cube inside a sphere to solve PDEs in cubical geometry using spectral bases, achieving moderately efficient and accurate solutions. For Rayleigh-Bénard convection in a cube, it provides the first scaling result of heat flux with thermal forcing.
A new gridding technique for the solution of partial differential equations in cubical geometry is presented. The method is based on volume penalization, allowing for the imposition of a cubical geometry inside of its circumscribing sphere. By choosing to embed the cube inside of the sphere, one obtains a discretization that is free of any sharp edges or corners. Taking full advantage of the simple geometry of the sphere, spectral bases based on spin-weighted spherical harmonics and Jacobi polynomials, which properly capture the regularity of scalar, vector and tensor components in spherical coordinates, can be applied to obtain moderately efficient and accurate numerical solutions of partial differential equations in the cube. This technique demonstrates the advantages of these bases over other methods for solving PDEs in spherical coordinates. We present results for a test case of incompressible hydrodynamics in cubical geometry: Rayleigh-Bénard convection with fully Dirichlet boundary conditions. Analysis of the simulations provides what is, to our knowledge, the first result on the scaling of the heat flux with the thermal forcing for this type of convection in a cube in a sphere.