Numerical solution of the initial boundary value problem with vacuum boundary conditions for the magnetic field induction equation in a ball

The aim of this work is to develop and test an algorithm for numerical solution of the initialboundary value problem with vacuum boundary conditions for the equation of magnetic field induction in a ball based on a modification of finite-difference methods of computational fluid dynamics and electrodynamics for use in orthogonal curvilinear coordinates.
The problem of the numerical solution of the magnetic field induction equation in the ball $\mathrm{G}$ is considered in this paper using the magnetohydrodynamics (MHD) model. It is assumed that outside the ball the magnetic field induction $\mathbf{B} = \mathrm{grad}\, \psi$, and the potential $\psi$ is a harmonic function regular at infinity. At the boundary $\partial G$, the vacuum boundary conditions are set. They include the requirement that the magnetic field is continuous and the normal component of the current density is equal to zero.
The discretization of the induction equation is carried out using the control-volume method and the FDTD (Finite Difference Time Domain) method modified for orthogonal curvilinear coordinates. When integrating over time, a completely implicit scheme is used. The total (convective and diffusion) flows are approximated according to the power-law scheme.
The proposed algorithm consists in sequentially executing the following steps at each time layer:
$\underline{\text{Step 1}}$: The approximation of the radial component of the magnetic field at $\partial G$ is determined from the discretized continuity equation for the boundary control volumes.
$\underline{\text{Step 2}}$: The external Neumann problem for the Laplace equation is solved using the Kelvin transformation. The approximation for the tangential components of the magnetic field at the $\partial G$ is calculated using the found potential $\psi$.
$\underline{\text{Step 3}}$: The found components of $\mathbf{B}$ are used as boundary conditions for finding a solution to the induction equation.
The steps 1 to 3 are repeated until convergence is achieved at a given moment in time.
The algorithm was tested on the problem of the magnetic field diffusion in a conducting ball, which has an analytical solution.
In this paper, the discretization of the magnetic field induction equation the solution of which satisfies the solenoidality condition is obtained in arbitrary orthogonal curvilinear coordinates. A new algorithm for numerical solution of the initial-boundary value problem with vacuum boundary conditions for the equation of magnetic field induction in a ball is developed. The comparison between the numerical solution and the analytical solution demonstrates the convergence of the constructed finite-difference scheme.