Solver comparison for Poisson-like equations on tokamak geometries

Abstract

The solution of Poisson-like equations defined on a complex geometry is required for gyrokinetic simulations, which are important for the modelling of plasma turbulence in nuclear fusion devices such as the ITER tokamak. In this paper, we compare three existing solvers finely tuned to solve this problem, in terms of the accuracy of the solution, and their computational efficiency. We also consider practical implementation aspects, including the parallel efficiency of the code, potentially enabling an integration of the solvers in a state-of-the-art first-principle gyrokinetic simulation framework. The first, the Spline FEM solver, uses C1 polar splines to construct a finite elements method which solves the equation on curvilinear coordinates. The resulting linear system is solved using a conjugate gradient method. The second, the GMGPolar solver, uses a symmetric finite difference method to discretise the differential equation. The resulting linear system is solved using a tailored geometric multigrid scheme, with a combination of zebra circle and radial line smoothers, together with an implicit extrapolation scheme. The third, the Embedded Boundary solver, uses a finite volumes method on Cartesian coordinates with an embedded boundary scheme. The resulting linear system is solved using a multigrid scheme. The Spline FEM solver is shown to be the most accurate. The GMGPolar solver is shown to use the least memory. The Embedded Boundary solver is shown to be the fastest in most cases. All three solvers are shown to be capable of solving the equation on a realistic non-analytical geometry. The Embedded Boundary solver is additionally used to attempt to solve an X-point geometry.