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

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 Vlasovlike 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. arXiv:1909.05005v1 [physics.comp-ph]

Energy-Minimizing, Symmetric Discretizations for Anisotropic Meshes and Energy Functional Extrapolation

“…2D curvilinear coordinates can also be used to describe the cross sections of more complicated three-dimensional geometries. An example motivating this method in this article in particular, is the Tokamak geometry used in plasma fusion [2,28].…”

“…where r 1 > 0; as introduced and then used in [2,28] to describe tokamak cross-sections from fusion plasma applications. According to [2,28], we use κ = 0.3 and δ = 0.2.…”

“…where r 1 > 0; as introduced and then used in [2,28] to describe tokamak cross-sections from fusion plasma applications. According to [2,28], we use κ = 0.3 and δ = 0.2. We also provide results for κ = δ = 0 which results in a standard polar coordinate transformation.…”

Implicitly Extrapolated Geometric Multigrid on Disk-Like Domains for the Gyrokinetic Poisson Equation from Fusion Plasma Applications

“…1, center) or more realistic D-shaped geometries were found to be advantageous; see, e.g., [5,10]. For more details on the problem setting and the physical details, we refer the reader to [5,10,13,32,[37][38][39]. Here, we propose a tailored solver for −∇ • (α∇u) = f in Ω, u = 0 on ∂Ω, (1.1) where Ω ⊂ R 2 is a disk-like domain, f : Ω → R, and α : Ω → R is a varying coefficient, also refered to as density profile.…”

“…The purpose of this article is to develop a problem-specific solver for this scenario. Our particular problem setting is taken from [5,13,32,38,39]. The solution of this system is a part of the iterative solution process in large gyrokinetic codes such as Gysela [13] where the 2D problem must be solved repeatedly over many times steps on hundreds or thousands of such 2D cross sections.…”