Super Time Stepping Methods for Diffusion using Discontinuous-Galerkin Spatial Discretizations

TL;DR

The paper tackles stiff diffusion terms in gyrokinetic simulations, where the diffusion operator can have a dominant negative real eigenvalue $\lambda$ and Jacobian assembly is impractical. It evaluates Jacobian-free STS methods (RKC, RKL) alongside DIRK and SSP schemes on both finite-difference and discontinuous Galerkin discretizations, introducing a DG-aware temporal error norm and automatic dominant-eigenvalue estimation to size STS stages. Results show STS approaches deliver superior efficiency for diffusion-dominated problems, with a cell-wise temporal norm further improving adaptivity by reducing step rejections; using $\lambda_{\text{approx}}$ for stage budgeting outperforms a conservative analytical $\lambda_{\text{user}}$. The work provides practical guidance for integrating STS within gyrokinetic codes and informs future operator-splitting strategies to couple diffusion with advection and multi-physics components in scalable, Jacobian-free time stepping.

Abstract

Super-time-stepping (STS) methods provide an attractive approach for enabling explicit time integration of parabolic operators, particularly in large-scale, higher-dimensional kinetic simulations where fully implicit schemes are impractical. In this work, we present an explicit STS framework tailored for diffusion operators in gyrokinetic models, motivated by the fact that constructing and storing a Jacobian is often infeasible due to strong nonlocal couplings, high dimensionality, and memory constraints. We investigate the performance of several STS methods, including Runge-Kutta-Chebyshev (RKC) and Runge-Kutta-Legendre (RKL) schemes, applied to a diffusion equation discretized using both discontinuous Galerkin (DG) and finite-difference methods. To support time adaptivity, we introduce a novel error norm designed to more accurately track temporal error arising from DG spatial discretizations, in which degrees of freedom contribute unevenly to the solution error. Finally, we assess the performance of an automatic eigenvalue estimation algorithm for determining the required number of STS stages and compare it against an analytical estimation formula.

Figures (7)

• Figure 1: Computational efficiency for each adaptive solver for various tolerances, grids, and diffusion coefficients.
• Figure 2: Sensitivity to dominant eigenvalue safety factor ($q_{\lambda}$)
• Figure 3: Error vs fixed step size ($h$) for varying diffusion coefficient $\nu_{v_\|}$
• Figure 4: Error vs relative tolerance (rtol) for the component-wise and cell-wise norms, equations \ref{['eq:norm1']} and \ref{['eq:norm2']} respectively, when using RKL and SSP4.
• Figure 5: Error vs computation times for the component-wise and cell-wise norms, equations \ref{['eq:norm1']} and \ref{['eq:norm2']} respectively, when using RKL or SSP4.