Keaton J. Burns

Calum S. Skene, Keaton J. Burns

We present a general and automated approach for computing model gradients for PDE solvers built on sparse spectral methods, and implement this capability in the widely used open-source Dedalus framework. We apply reverse-mode automatic differentiation to symbolic graph representations of PDEs, efficiently constructing adjoint solvers that retain the speed and memory efficiency of this important class of modern numerical methods. This approach enables users to compute gradients and perform optimization for a wide range of time-dependent and nonlinear systems without writing additional code. The framework supports a broad class of equations, geometries, and boundary conditions, and runs efficiently in parallel using MPI. We demonstrate the flexibility and capabilities of this system using canonical problems from the literature, showing both strong performance and practical utility for a wide variety of inverse problems. By integrating automatic adjoints into a flexible high-level solver, our approach enables researchers to perform gradient-based optimization and sensitivity analyses in spectral simulations with ease and efficiency.

Keaton J. Burns, Daniel Lecoanet, Geoffrey M. Vasil et al.

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.

Geoffrey M. Vasil, Keaton J. Burns, Daniel Lecoanet et al.

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r=0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.