A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation

Abstract

Please cite this article in press as: W.T. Taitano et al., A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth-Fokker-Planck equation, J. Comput. Phys. (2015), http://dx. AbstractIn this study, we demonstrate a fully implicit algorithm for the multi-species, multidimensional RosenbluthFokker-Planck equation which is exactly mass-, momentum-, and energy-conserving, and which preserves positivity. Unlike most earlier studies, we base our development on the Rosenbluth (rather than Landau) form of the Fokker-Planck collision operator, which reduces complexity while allowing for an optimal fully implicit treatment. Our discrete conservation strategy employs nonlinear constraints that force the continuum symmetries of the collision operator to be satisfied upon discretization. We converge the resulting nonlinear system iteratively using Jacobian-free Newton-Krylov methods, effectively preconditioned with multigrid methods for efficiency. Single-and multi-species numerical examples demonstrate the advertised accuracy properties of the scheme, and the superior algorithmic performance of our approach. In particular, the discretization approach is numerically shown to be second-order accurate in time and velocity space and and to exhibit manifestly postive entropy production. That is, H-theorem behavior is indicated for all the examples we have tested. The solution approach is demonstrated to scale optimally with respect to grid refinement (with CPU time growing linearly with the number of mesh points), and timestep (showing very weak dependence of CPU time with time-step size). As a result, the proposed algorithm delivers several orders-of-magnitude speedup vs. explicit algorithms.

Conservative discrete-velocity method for the ellipsoidal Fokker-Planck equation in gas-kinetic theory

“…This problem is addressed in several works in the topic of FPL equation [6,31]. For RFP equation which has a similar mathematical form as the FP and ES-FP equations used in gas-kinetic theory, Ref [19] constructed a conservative finite volume scheme in velocity space.…”

“…By using a finite volume framework in velocity space and extra coefficients on advection terms for conservation purpose, RFP equation is well solved by the deterministic numerical method in Ref. [19,20].…”

Differential formulation of the gyrokinetic Landau operator

“…In the case of axially symmetric particle phase-space operator or, equivalently, in the zero-Larmor-radius limit of the gyrokinetic operator, conservative numerical methods have been developed for both the Landau and the potential formulations (see e.g. Hager et al (2016); Taitano et al (2015)). On the other hand, it has been explicitly shown in Burby et al (2015) that energetically consistent collisional gyrokinetics requires both the Vlasov and the collision operator to be treated equally at the same order with respect to the asymptotic gyrocentre transformation.…”

Positivity-preserving scheme for two-dimensional advection–diffusion equations including mixed derivatives

“…Two dimensional advection-diffusion equations have widespread applications in physics, engineering and finance, and can generally be written as u t = Au xx + Bu xy + Cu yy + Du x + Eu y + F u (1) where u = u(x, y, t). As these equations are often too difficult to solve analytically, numerical solutions are required.…”

“…In this paper, we propose a scheme for improving the accuracy of lower-order methods, in particular with respect to the preservation of positivity, when solving two-dimensional advection-diffusion equations in the presence of mixed derivatives. The proposed scheme is applicable to any advection-diffusion equation of the form (2), as well as equations of the form (1) where the solution must remain positive. It is applied to a two-dimensional diffusion equation with mixed derivatives, for which the analytical solution is known, as well as the Fokker-Planck collision operator in cylindrical coordinates.…”