Local high order space-time apdaptive MLSDC

Abstract: 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.

• The paper introduces a local high-order space-time adaptive MLSDC method that combines DG-SEM with semi-implicit MLSDC to solve nonlinear conservation laws.
• It employs a novel error estimator that integrates spatial spectral decay and temporal extrapolation to drive local h/p and p adaptivity.
• Numerical experiments on Burgers’ and Euler equations demonstrate substantial runtime savings and improved accuracy compared to uniform discretizations.

High-Order Local Space-Time Adaptive MLSDC for Conservation Laws

Overview and Context

The paper "Local high order space-time apdaptive MLSDC" (2606.14369) introduces a highly technical hybridization of discontinuous Galerkin spectral element space discretization with semi-implicit Multilevel Spectral Deferred Correction (MLSDC) time integration, equipped with genuine local space-time adaptivity. This framework incorporates arbitrary-order $h$- and $p$-refinement in space, together with multilevel $p$-refinement in time, and leverages a novel error estimator to trigger local adaptivity. The method targets nonlinear conservation laws, with numerical experiments spanning Burgers' equation and the Euler equations, including classical benchmark cases.

Methodological Foundations

The method builds on semi-implicit MLSDC [MG_Pfister2025], extending it to local adaptive refinement. The space-time discretization employs DG-SEM, partitioning the domain into spectral elements (spatial) and time slabs (temporal), with polynomial representations in both. The MLSDC framework iteratively solves collocation equations at Radau-Right nodes, facilitating high-order accuracy ($2M_l-1$ in time for $M_l$ nodes).

Figure 1: Multilevel space-time discretization illustration; level $l+1$ exemplifies increased spatial resolution ($h$-refinement) and temporal nodes ($p$-refinement) over level $l$. Orange points mark space-time coefficients $Q_l^{e,n}$.

A genuinely space-time adaptive mesh routine is introduced. Elements are categorized as active (twig or leaf), frozen (interpolated from coarser levels), with refinement and activity markers managed by the error estimator. Refinement in time is restricted to $p$0-adjustment (collocation nodes), while spatial refinement admits arbitrary $p$1/$p$2.

Figure 2: Adaptivity workflow: local $p$3/$p$4 refinement in space, $p$5 refinement in time, element classes, and mesh evolution.

Transfer operators (interpolation, restriction, projection) are realized as algebraic matrices acting on the coefficient vectors; fine-to-coarse mechanisms aggregate residuals and prolongate corrections, strictly within active regions.

Brandt's FAS multigrid paradigm is invoked: on each adaptive level, collocation equations are solved on active elements, with FAS right-hand side contributions (computed from finer level projections and restricted residuals) present only on non-finest levels. The adaptive MLSDC algorithm integrates pre-/post-smoothing, correction prolongation, and FAS right-hand side assembly, strictly restricted to active mesh regions.

Error Estimation and Adaptivity

A key innovation is a unified local space-time error estimator. For spatial error, the spectral estimator [EE_MavriplisDiss1989] models truncation and quadrature/interpolation error via the modal decay of Legendre coefficients, extended and fit across the polynomial basis. The temporal estimator, novel to this work, constructs a dual-part indicator: (i) extrapolation error from difference between successive SDC iterates at the final collocation node (Richardson-style), and (ii) a superconvergent truncation error leveraging SDC's endpoint accuracy ($p$6), integrating spectral coefficients beyond this order.

Figure 3: Process for spatial truncation error estimation at the final temporal node $p$7.

Both estimators are individually calibrated and filtered for numerical noise. Their sum—conservative by construction—guides local mesh refinement; the estimator is validated against measured $p$8 error in controlled experiments, consistently providing a reliable refinement indication.

Figure 4: Spatially dominated error regime—comparison of estimated (overpredictive) vs actual error across polynomial orders.

Numerical Analysis and Benchmark Results

Burgers' Equation Moving Front

The adaptive MLSDC$p$9 method is applied to the Burgers’ equation with a moving shock front ($p$0, $p$1). Adaptive refinement, governed by stringent thresholding of the error estimator, dynamically tracks the propagation of the shock, refining only where needed and coarsening in its wake.

Figure 5: Solution of Burgers' moving front at $p$2, element boundaries visualize refinement locally around the shock.

Comparisons show that runtime is reduced to $p$3 of that required by an equivalent globally refined discretization. Error-runtime diagrams establish that adaptive MLSDC outperforms both classical TVD-RK3 and single-level SDC/MLSDC at the same accuracy, with increased proportional gains as error thresholds decrease.

Figure 6: Error versus runtime for IMEX RK, SDC, globally and locally adaptive MLSDC on Burgers’ moving front ($p$4).

Refinement Criteria Comparative Study

The space-time error estimator is contrasted with alternative criteria (exact $p$5 error, Henderson’s indicator [EE_Henderson1999], Persson’s artificial diffusivity [SE_Persson2006a]). The spectral estimator and exact error lead to highly localized refinement zones and the lowest runtimes at fixed accuracy; artificial diffusivity over-refines and is suboptimal; Henderson’s estimator proves quantitatively efficient, but fails on tests with discontinuities.

Shu-Osher Shock-Fluctuation Benchmark

For the 1D Euler Shu-Osher problem [TI_Shu1989a], adaptive MLSDC$p$6 achieves the accuracy of the finest uniform SDC discretization while incurring $p$7 lower computational cost. The error estimator robustly captures both the shock front and oscillatory regions, ensuring compact, dynamically evolving refinement regions.

Figure 7: Solution and local error estimate for Shu-Osher shock-fluctuation at $p$8, element boundaries indicate local mesh adaptivity.

Diagnostic figures demonstrate that coarse and intermediate uniform SDC discretizations significantly underresolve the solution, while adaptive MLSDC achieves parity with the finest level.

Figure 8: Solution for single-level SDC (level 1), highlighting underresolved features.

Figure 9: Difference to reference for single-level SDC (level 1), visualizing local error misdistribution.

Theoretical and Practical Implications

The method demonstrates that genuine local space-time adaptive refinement can be harnessed in high-order DG-SEM/MLSDC frameworks, yielding:

• Compact localized refinement regions (space and time), directly linked to the estimated local discretization error.
• Arbitrarily high-order accuracy in both space and time ($p$9-th order achievable), exceeding historic limits of four for prior adaptive MLSDC methods.
• Robust tracking of moving discontinuities and oscillatory features, leveraging spectral error estimators.
• Significant reduction in computational cost, with efficiency gains scaling with problem dimensionality.

The inclusion of a spectral estimator for time—sensitive to SDC collocation superconvergence—is a pronounced technical advancement, enabling error-driven $2M_l-1$0-refinement in time.

Future Directions

The work outlines several promising trajectories:

• Extension to three spatial dimensions, with anticipated multiplicative efficiency gains from localized refinement.
• Exploration of anisotropic adaptivity: independently triggering spatial or temporal refinement based on dominant error—made possible by the additive error structure.
• Introduction of robust temporal $2M_l-1$1-adaptivity protocols and assessment of MLSDC schemes based on alternative Runge-Kutta collocations with improved stability properties (e.g., TVD, SSP), enhancing performance on convection-dominated regimes.
• Systematic optimization of refinement thresholds and adaptivity criteria, potentially informed by adjoint-based or output-oriented indicators.

Conclusion

This paper provides a comprehensive framework for local high-order space-time adaptive simulation of conservation laws using MLSDC and DG-SEM discretizations. The method retains full arbitrary-order accuracy, robust conservation, and efficient local adaptivity in both space and time, validated on nonlinear scalar and systems test problems, and demonstrates substantial practical computational savings. Future research will likely focus on high-dimensional deployments, anisotropic adaptivity, and further refinement of error estimation and adaptivity protocols.