A novel efficient solver for Ampere's equation in general toroidal topologies based on singular value decomposition techniques

• Published in: Journal of computational physics Vol. 406; p. 109214
• Main Authors: Reynolds-Barredo, J.M., Peraza-Rodríguez, H., Sanchez, R., Tribaldos, V.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01-04-2020; Elsevier Science Ltd
• Subjects: Ampere's law; Boundary conditions; Computational physics; Divergence; Finite difference method; Kernels; Magnetic fields; Magnetic vector potentials; MHD; Operators (mathematics); Singular value decomposition; Spectral method; SVD; Topology; Toroidal topology; SVD; Toroidal topology; MHD; Spectral method; Ampere's law; Singular value decomposition
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.109214

•A new fast method is proposed to solve Ampere's equation in toroidal domains.•The method uses a mixed spectral – difference discretization.•A fast solver is found using the block structure of the involved matrices and SVD decomposition techniques.•The solution magnetic field has a zero divergence up to machine precision.•A successful comparison with an integral method is carried out for the case of W7-X stellarator.
A new method is proposed to solve Ampere's equation in an arbitrary toroidal domain in which all currents are known, given proper boundary conditions for the magnetic vector potential. The novelty of the approach lies in the application of singular value decomposition (SVD) techniques to tackle the difficulties caused by the kernel associated by the curl operator. This kernel originates physically due to the magnetic field gauge. To increase the efficiency of the solver, the problem is represented by means of a dual finite difference-spectral scheme in arbitrary generalized toroidal coordinates, which permits to take advantage of the block structure exhibited by the matrices that describe the discretized problem. The result is a fast and efficient solver, up to three times faster than the double-curl method in some cases, that provides an accurate solution of the differential form of Ampere law while guaranteeing a zero divergence of the resulting magnetic field down to machine precision.