The "Sphered Cube": A New Method for the Solution of Partial Differential Equations in Cubical Geometry

Understand

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.