A second-order accurate Super TimeStepping formulation for anisotropic thermal conduction

Abstract

Astrophysical fluid dynamical problems rely on efficient numerical solution techniques for hyperbolic and parabolic terms. Efficient techniques are available for treating the hyperbolic terms. Parabolic terms, when present, can dominate the time for evaluating the solution, especially when large meshes are used. This stems from the fact that the explicit time‐step for parabolic terms is proportional to the square of the mesh size and can become unusually small when the mesh is large. Multigrid‐Newton–Krylov methods can help, but usually require a large number of iterations to converge. Super TimeStepping schemes are an interesting alternative, because they permit one to take very large overall time‐steps for the parabolic terms while using only a modest number of explicit time‐steps. Super TimeStepping schemes of the type used in astrophysics have, so far, been only first‐order accurate in time and prone to instabilities. In this paper, we present a Runge–Kutta method that is based on the recursion sequence for Legendre polynomials, called the RKL2 method. RKL2 is a time‐explicit method that permits us to treat non‐linear parabolic terms robustly and with large, second‐order accurate time‐steps. An s‐stage RKL2 scheme permits us to take a time‐step that is ∼s2 times larger than a single explicit, forward Euler time‐step for the parabolic operator. This permits an s‐fold gain in computational efficiency over explicit time‐step sub‐cycling. For modest values of ‘s’, the advantage can be substantial. The stability properties of the new schemes are explored and they are shown to be stable and positivity preserving for linear operators. We document the method as it is applied to the anisotropic thermal conduction operator for dilute, magnetized, astrophysical plasmas. Implementation‐related details are discussed. The RKL2 Super TimeStepping scheme has been implemented in the riemann code for computational astrophysics. We explain the method for picking an s‐stage RKL2 scheme for the parabolic terms and show how it can be integrated with a hyperbolic system solver. The method’s simplicity makes it very easy to retrofit the s‐stage RKL2 scheme to any problem with a parabolic part when a well‐formed spatial discretization is available. Several stringent test problems involving thermal conduction in astrophysical plasmas are presented and the method is shown to perform robustly and efficiently on all of them.

Scalable explicit implementation of anisotropic diffusion with Runge–Kutta–Legendre super-time stepping

“…This Appendix is closely based on Meyer et al (2012Meyer et al ( , 2014 and is included here for completeness. Second order accuracy for the RKL-2 scheme can be achieved by matching the first three terms in Eqs.…”

“…We consider an MHD blast wave in cylindrical co-ordinates with initial parameters similar to the L1 model of Meyer et al (2012) but without radiative cooling. For this 2D axisymmetric test, we solve the standard set of ideal MHD equations, taking into account anisotropy in thermal conduction flux along with saturated conduction with φ = 0.3 (see Eqs.…”

“…Recently, Meyer et al (2012Meyer et al ( , 2014 have proposed Legendre polynomials (LPs) as the basis for the construction of robust STS schemes. For a general s-stage RKL scheme, the stability polynomial is chosen to be of the form…”

“…A variant of this method was popularized by Alexiades et al 1996 (AAG-STS). More recently, super-time-stepping based on Legendre polynomials, known as Runge-Kutta Legendre STS (RKL-STS), was proposed by Meyer et al (2012Meyer et al ( , 2014.…”

“…Treating parabolic terms with accelerated super-time stepping (STS) methods has been discussed in literature but these methods suffer from poor accuracy (first order in time) and also have difficult-to-choose tuneable stability parameters. In this work we highlight a second order (in time) Runge Kutta Legendre (RKL) scheme (first described by Meyer et al 2012) that is robust, fast and accurate in treating parabolic terms alongside the hyperbolic conversation laws. We demonstrate its superiority over the first order super time stepping schemes with standard tests and astrophysical applications.…”