Robust semi-implicit multilevel SDC methods for conservation laws

TL;DR

This work introduces a robust semi-implicit multilevel spectral deferred correction (SI-MLSDC) framework for conservation laws, combining semi-implicit Lax-Wendroff–style time integrators with multilevel corrections and high-order DG-SEM spatial discretization. Dahlquist-type stability analysis shows SI(2) offers superior stability and accuracy, while MLSDC substantially reduces fine-grid iterations compared to single-level SDC, across convection–diffusion, Burgers, Euler, and Navier–Stokes problems. The study also highlights the non-equivalence of incremental and non-incremental MLSDC formulations and evaluates projection strategies (embedded interpolation vs. L^2 projection) and starting strategies (cascade, FMG) for performance. Limitations arise with under-resolved coarse grids and high CFLs, but the results indicate strong potential for space-time p-refinement and future extension to shock-capturing and three-dimensional problems with adaptive strategies. Overall, SI-MLSDC offers a scalable, high-order, space-time framework with practical stability and efficiency benefits for complex conservation-law simulations.

Abstract

Semi-implicit multilevel spectral deferred correction (SI-MLSDC) methods provide a promising approach for high-order time integration for nonlinear evolution equations including conservation laws. However, existing methods lack robustness and often do not achieve the expected advantage over single-level SDC. This work adopts the novel SI time integrators from [48] for enhanced stability and extends the single-level SI-SDC method with a multilevel approach to increase computational efficiency. The favourable properties of the resulting SI-MLSDC method are shown by linear temporal stability analysis for a convection-diffusion problem. The robustness and efficiency of the fully discrete method involving a high-order discontinuous Galerkin SEM discretization are demonstrated through numerical experiments for the convection-diffusion, Burgers, Euler and Navier-Stokes equations. The method is shown to yield substantial reductions in fine-grid iterations compared to single-level SI-SDC across a broad range of test cases. Finally, current limitations of the SI-MLSDC framework are identified and discussed, providing guidance for future improvements.

Figures (18)

• Figure 1: Sketches for the starting strategies Cascade and FMG$_1$ , i.e. one cycle per added level, each with one attached full V-cycle in blue
• Figure 2: Neutral stability curves for incremental MLSDC methods with Radau Right points and different time integrators shown after each odd V-cycle. FMG$_1$ denotes the full multigrid start strategy (algorithm \ref{['alg:mlsdc:fmg']}) with one cycle per added level, i.e., $C_{\text{\rmfamily\scshape fmg}}=1$
• Figure 3: Neutral stability curves for non-incremental MLSDC methods with Radau Right points and different time integrators shown after each odd V-cycle
• Figure 4: Accuracy curves ($\varepsilon = 10^{-6}$) for MLSDC methods with Radau Right points and different time integrators shown after each odd V-cycle for the incremental and non-incremental formulation. FMG$_1$ denotes the full multigrid start strategy (algorithm \ref{['alg:mlsdc:fmg']}) with one cycle per added level, i.e., $C_{\text{\rmfamily\scshape fmg}}=1$
• Figure 5: MLSDC V-cycle vs. single-level SDC corrector, number of coarse sweeps $N_{c} = 2$, predictor variant

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Proposition 1
• Remark 8
• Remark 9