# Thomas Adams Masters Project

Algorithms for Computing the Inverse of a Curl

Abstract: In order to transform data from a numerical simulation of binary black holes in which only the magnetic field B is computed, to another simulation in which only the vector potential A is computed, ∇ × A = B must be solved to generate a vector potential from the known magnetic field vector. We implement multiple methods for solving this equation based on finite difference numerical representations of the equation. One method of solving this equation directly is to solve the problem sequentially point-by-point, using the solution at previous points as constraints.

Two other methods of solving this problem focus on taking the curl of both sides of the equation and solve the resulting Poisson's equation. The first of these methods is to set up a large but very sparse matrix and solve globally using Gaussian Elimination. The second of these methods is perform a multigrid solve by setting up the problem on grids of multiple resolutions and solving on each level using an iterative technique.

Of the two Poisson's equation methods the multigrid solver is determined through testing to have superior computational scaling for very large magnetic fields to the global solver. It also holds the advantage over the direct point-by-point method of being able to easily incorporate adaptive mesh refinement which is needed for solving the problem on the incredibly large magnetic fields used by the simulations. This leads to the conclusion that the multigrid method should be the primary focus for testing and implementation.

Date: 7/12/2017

Time: 3:30PM-4:30PM

Place: 415 Hodges

All are welcome.