Local time-integration for Friedrichs' systems

TL;DR

This work develops a fully explicit local time-stepping scheme for Friedrichs' two-field systems discretized by discontinuous Galerkin methods on locally refined meshes, addressing stiffness arising from small, fast regions. By combining a leapfrog time integrator on the nonstiff part with a filter—constructed from a scaled and shifted Chebyshev polynomial—applied to the stiff component, the method achieves a CFL condition that is largely independent of the stiff region while keeping the computational overhead modest. The authors prove stability and convergence for the entire class of local time-integration schemes under suitable filter parameters, showing second-order accuracy in time and optimal order in space, with LI and certain LTI variants recovered as special cases. Numerical experiments with Maxwell equations confirm the predicted convergence rates and demonstrate substantial runtime savings of the LFC-LTS method over both leapfrog and LI approaches, highlighting practical impact for efficiently simulating wave-type problems on locally refined meshes.

Abstract

In this paper, we address the full discretization of Friedrichs' systems with a two-field structure, such as Maxwell's equations or the acoustic wave equation in div-grad form, cf. [14]. We focus on a discontinuous Galerkin space discretization applied to a locally refined mesh or a small region with high wave speed. This results in a stiff system of ordinary differential equations, where the stiffness is mainly caused by a small region of the spatial mesh. When using explicit time-integration schemes, the time step size is severely restricted by a few spatial elements, leading to a loss of efficiency. As a remedy, we propose and analyze a general leapfrog-based scheme which is motivated by [5]. The new, fully explicit, local time-integration method filters the stiff part of the system in such a way that its CFL condition is significantly weaker than that of the leapfrog scheme while its computational cost is only slightly larger. For this scheme, the filter function is a suitably scaled and shifted Chebyshev polynomial. While our main interest is in explicit local-time stepping schemes, the filter functions can be much more general, for instance, a certain rational function leads to the locally implicit method, proposed and analyzed in [24]. Our analysis provides sufficient conditions on the filter function to ensure full order of convergence in space and second order in time for the whole class of local time-integration schemes.

Figures (3)

• Figure 1: Locally refined mesh of the domain $(0,1)^2$.
• Figure 2: Error of the numerical solution of \ref{['eq:TE-system']} with initial data given by \ref{['eq:TE-exact-sol']} obtained by the leapfrog method (blue) and the local time-stepping method \ref{['eq:lts-scheme']} with filter \ref{['eq:lfc-polynomials']}, polynomial degrees $p = 2$ (orange), $p= 4$ (green), $p = 8$ (red), $p = 9$ (purple), and stabilization \ref{['eq:special-stab-param']} with $\eta = 1$. The space discretization is done with dG degree $k = 5$ and a three times at the center locally refined mesh, see \ref{['fig:2D-mesh']}. The dash-dotted line depicts the maximal stable time stepsize of leapfrog method used on the coarse mesh $\mathcal{T}_{h^{},\mathrm{lf}}$.
• Figure 3: Error of the LFC-LTS method for the example in Section \ref{['subsec:stabilization-example']} with polynomial degrees $p = 3, 4, 5$. Left: without stabilization ($\eta = 0$), right: with stabilization ($\eta = 0.1$). The dash-dotted line depicts the maximal stable time-stepsize of the leapfrog method on the coarse mesh $\mathcal{T}_{h^{},\mathrm{lf}}$.

Theorems & Definitions (27)

• Theorem 3.1
• Lemma 3.2
• Definition 3.3
• Definition 3.4
• Lemma 3.5
• proof
• Lemma 3.6
• proof
• Lemma 3.7
• proof