Solving hyperbolic-elliptic problems on singular mapped disk-like domains with the method of characteristics and spline finite elements

TL;DR

This paper develops robust numerical methods combining characteristic-based semi-Lagrangian advection and spline finite elements to solve hyperbolic-elliptic PDEs on singular disk-like domains, effectively handling coordinate singularities.

Contribution

It introduces a comprehensive numerical strategy using pseudo-Cartesian coordinates and high-order spline finite elements for hyperbolic-elliptic problems on singular geometries, including point charges.

Findings

High-order convergence achieved across the domain

Methods effectively handle singularities at the pole

Applicable to particle-in-cell simulations with point charges

Abstract

A common strategy in the numerical solution of partial differential equations is to define a uniform discretization of a tensor-product multi-dimensional logical domain, which is mapped to a physical domain through a given coordinate transformation. By extending this concept to a multi-patch setting, simple and efficient numerical algorithms can be employed on relatively complex geometries. The main drawback of such an approach is the inherent difficulty in dealing with singularities of the coordinate transformation. This work suggests a comprehensive numerical strategy for the common situation of disk-like domains with a singularity at a unique pole, where one edge of the rectangular logical domain collapses to one point of the physical domain (for example, a circle). We present robust numerical methods for the solution of Vlasov-like hyperbolic equations coupled to Poisson-like elliptic equations in such geometries. We describe a semi-Lagrangian advection solver that employs a novel set of coordinates, named pseudo-Cartesian coordinates, to integrate the characteristic equations in the whole domain, including the pole, and a finite element elliptic solver based on globally $C^1$ smooth splines (Toshniwal et al., 2017). The two solvers are tested both independently and on a coupled model, namely the 2D guiding-center model for magnetized plasmas, equivalent to a vorticity model for incompressible inviscid Euler fluids. The numerical methods presented show high-order convergence in the space discretization parameters, uniformly across the computational domain, without effects of order reduction due to the singularity. Dedicated tests show that the numerical techniques described can be applied straightforwardly also in the presence of point charges (equivalently, point-like vortices), within the context of particle-in-cell methods.