Task-Based Parallelization of an Implicit Kinetic Scheme

Abstract

In this paper we present and implement the Palindromic Discontinuous Galerkin (PDG) method in dimensions higher than one. The method has already been exposed and tested in [4] in the one-dimensional context. The PDG method is a general implicit high order method for approximating systems of conservation laws. It relies on a kinetic interpretation of the conservation laws containing stiff relaxation terms. The kinetic system is approximated with an asymptotic-preserving high order DG method. We describe the parallel implementation of the method, based on the StarPU runtime library. Then we apply it on preliminary test cases.

High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation

“…We have not yet compared the efficiency of our approach with other explicit or implicit DG solvers. However, we have observed a good parallel scaling of the method when the number of computational cores increases [5]. In addition, as it is shown in the numerical sections, the PDG method accepts very high CFL numbers, which makes it a good candidate for avoiding costly non-linear implicit solvers.…”

“…In addition, because the free transport step is solved by an upwind DG solver, then the linear operator Id + ∆t L h is block-triangular [5] and its inverse T 1 can also be computed explicitly. We detail the method in Section 4.…”

“…Those local sparse linear systems are solved with the KLU library, which is able to detect efficiently block-triangular structures [15]. More details on the implementation are given in [5]. For the moment, the local systems are assembled and factorized at each timestep.…”

“…See for instance in [6,42,12,35,32]. In a recent work we have evaluated the parallel scalability of the triangular solver [5]. The second fundamental aspect of our method is the construction of a symmetricin-time integrator that remains second order accurate even for vanishing relaxation time τ (AP property).…”

A robust and efficient solver based on kinetic schemes for Magnetohydrodynamics (MHD) equations

“…It leads to natural numerical methods, where the transport step and the collision step are made separately. Other interesting features arise from this representation (see [2,11], for instance).…”