Erik Pfister | Searcharxiv

arXiv2026

Local high order space-time adaptive MLSDC

Building upon the semi-implicit multilevel spectral deferred correction (SI-MLSDC) method introduced by Pfister and Stiller [38], this work presents a 1D space-time adaptive high-order method combining discontinuous Galerkin spectral element discretizations with multilevel spectral deferred corrections. The proposed approach enables genuine arbitrary-order accuracy in space and time while dynamically balancing spatial and temporal discretization errors to reduce computational cost. A key contribution is the development of a novel temporal error estimator that in combination with a spectral error estimator in space provides a reliable basis for adaptive refinement decisions. The error estimator is compared with two alternative refinement criteria to assess their impact on accuracy, computational efficiency and their suitability for complex problems. The performance of the adaptive method is demonstrated for nonlinear conservation laws ranging from Burgers' equation to the Euler equations. Numerical results show substantial runtime reductions while maintaining the desired accuracy. In particular, significant computational savings are achieved for Burgers' equation, and challenging benchmark problems such as the Shu-Osher shock-fluctuation benchmark. These results demonstrate the potential of adaptive SI-MLSDC methods for efficient high-order space-time adaptive simulations of complex flow problems.

math.NA↗

arXiv2025

Robust semi-implicit multilevel SDC methods for conservation laws

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.

math.NA↗