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Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

Olindo Zanotti, Francesco Fambri, Michael Dumbser, Arturo Hidalgo

arXiv:1412.0081v2math.NAastro-ph.IMcs.NA

TL;DR

DG schemes can oscillate at discontinuities, and the paper extends a posteriori sub-cell ADER-WENO limiting to space-time adaptive DG meshes. The scheme retains high-order convergence in tests and resolves fine-scale flow features.

  • Problem

    DG schemes can produce undesirable oscillations at discontinuities, motivating a limiter that controls them while retaining high-order resolution.

    Source: §1 · p. 2

  • Method

    The paper combines ADER-DG with a posteriori ADER-WENO sub-cell recomputation and space-time AMR in two and three dimensions.

    Source: §1 · p. 3, §7 · p. 29

  • Results

    The scheme matches nominal convergence orders through N = 8 with AMR and local time stepping, while test cases report fine-scale flow structure and agreement between AMR and uniform-grid MHD results.

    Source: §6.1.1 · p. 12, §6.1.5 · p. 21, §6.2.1 · p. 27

  • Takeaways & Limitations

    The authors conclude that the method may contribute to computational fluid dynamics applications needing high resolution and low numerical dissipation.

    Source: §7 · p. 31

    Extension to equations with stiff source terms and true physical dissipative processes remains future work.

    Source: §7 · p. 31

Abstract

from arXiv · show

In this paper we present a novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space-time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order \aposteriori sub-cell ADER-WENO finite volume \emph{limiter}. Notoriously, the original DG method produces strong oscillations in the presence of discontinuous solutions and several types of limiters have been introduced over the years to cope with this problem. Following the innovative idea recently proposed in \cite{Dumbser2014}, the discrete solution within the troubled cells is \textit{recomputed} by scattering the DG polynomial at the previous time step onto a suitable number of sub-cells along each direction. Relying on the robustness of classical finite volume WENO schemes, the sub-cell averages are recomputed and then gathered back into the DG polynomials over the main grid. In this paper this approach is implemented for the first time within a space-time adaptive AMR framework in two and three space dimensions, after assuring the proper averaging and projection between sub-cells that belong to different levels of refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. The spectacular resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, both for the Euler equations of compressible gas dynamics and for the magnetohydrodynamics (MHD) equations.

1. Introduction

DG methods are effective in smooth regions but oscillate near discontinuities, while high-order time integration and robust limiting remain important design challenges. This work combines ADER-DG, a posteriori sub-cell limiting, and space-time AMR.

Time integration

  • ADER time integration targets arbitrary order in space and time in one step, using an approximate Generalised Riemann Problem solution at element interfaces.The later local space-time DG predictor replaces the Cauchy–Kowalewski procedure and can handle stiff source terms, though explicit ADER schemes retain the CFL restriction.Source: §1 · p. 2

Motivation

  • DG schemes oscillate near sharp gradients and shocks, motivating limiters that address the Gibbs phenomenon while retaining high-order accuracy.The cited alternatives include artificial viscosity, spectral filtering, WENO limiting, and slope or moment limiting.Source: §1 · p. 2

A posteriori limiting

  • The earlier a posteriori DG limiter detects troubled cells, then recomputes them with ADER-WENO finite volume on (2N + 1)^d sub-cells chosen to match the DG time step.This approach is related to the MOOD paradigm.Source: §1 · p. 3

Contribution

  • The proposed space-time adaptive ADER-DG scheme combines arbitrary high-order accuracy, time-accurate local time stepping, and a posteriori sub-cell limiting without Runge–Kutta sub-steps or artificial viscosity.The introduction frames this as an improvement over existing DG-AMR approaches.Source: §1 · p. 3
  • Prior DG-AMR work often treats adaptivity as hp-adaptivity, whereas this paper aims to combine DG with space-time AMR and a proper sub-cell limiter.The introduction also situates AMR developments in finite-volume and DG methods.Source: §1 · p. 3

2. Mathematical framework: an overview

The framework represents hyperbolic conservation laws on a Cartesian mesh and advances a polynomial DG solution through a predictor, a one-step corrector, and an a posteriori sub-cell limiter. AMR refines or recoarsens the main grid, while the limiter addresses discontinuities.

Mathematical framework

  • The governing equations are nonlinear hyperbolic systems in conservative form, with conserved state u and nonlinear flux tensor F(u).The computational domain is discretized by a Cartesian grid.Source: §2 · p. 3

Method overview

  • The method combines a local space-time predictor, a one-step DG corrector, an a posteriori ADER-WENO finite-volume sub-cell limiter, and cell-by-cell AMR.AMR requires proper nesting within the sub-cell approach.Source: §2 · p. 4

AMR and limiting

  • AMR refines or recoarsens the main grid to follow physical features; the sub-cell limiter is used separately to handle shock waves and other discontinuities.The text emphasizes that main-grid adaptivity is not the same as sub-cell limiting.Source: §2 · p. 4

3. The ADER-DG scheme

ADER-DG first constructs a local space-time predictor from the DG polynomial, then uses a fully discrete one-step corrector to account for intercell fluxes. The method is order N + 1 in space and time for smooth solutions.

3.1. The local space-time predictor

  • The local predictor evolves the DG polynomial’s known spatial derivatives independently within each cell to approximate the generalized or derivative Riemann problem.In this DG formulation, no spatial reconstruction is needed.Source: §3.1 · pp. 4–5
  • The predictor’s nonlinear system solves for space-time expansion coefficients from the DG degrees of freedom at the previous time level, and can cope with stiff source terms.Stiff sources are absent from the systems considered in this paper.Source: §3.1 · p. 5

3.2. The fully discrete one-step ADER-DG scheme

  • The one-step corrector accounts for neighboring-cell fluxes omitted by the local predictor, using a Riemann solver for discontinuous interface states and the predictor for time-accurate flux integration.The method obtains the numerical flux from the left and right predictor states.Source: §3.2 · p. 6
  • For smooth solutions, the ADER-DG scheme achieves order N + 1 in both space and time, while unlimited DG fails at discontinuities because of Gibbs oscillations.In multiple dimensions, explicit DG time steps are constrained by the CFL condition, with a factor 2N + 1 in its denominator.Source: §3.2 · p. 6

4. A posteriori sub-cell limiter

The a posteriori limiter tests the DG candidate against physical positivity and a relaxed polynomial maximum principle. Troubled cells are recomputed on sub-cells with ADER-WENO finite volume, then projected back to the main grid.

Detection

  • The relaxed polynomial maximum principle checks the candidate against prior-step polynomial bounds over the cell and its Voronoi neighbors.Its relaxation permits small undershoots and overshoots and can avoid roundoff-error problems.Source: §4 · p. 7
  • A cell is marked troubled when either the physical positivity test or the relaxed discrete maximum principle fails; otherwise it is accepted.The positivity constraints are physically motivated, while the maximum-principle test is numerical.Source: §4 · p. 7

Sub-cell representation

  • The limiter initializes sub-cell averages by L2 projection of the previous DG solution onto (2N + 1)^d sub-cells.This sub-cell count matches the maximum time step of the main-grid DG and sub-grid finite-volume schemes, allowing the latter to use its maximum admissible CFL number.Source: §4 · p. 7

Recomputation and reconstruction

  • The troubled-cell solution is recomputed from the previous-time-level data with a robust third-order ADER-WENO finite-volume scheme, then reconstructed on the main grid.Using one-step schemes for both DG and sub-grid updates reduces total MPI communications relative to traditional Runge–Kutta schemes.Source: §4 · pp. 7–8

5. AMR with the sub-cell limiter

The section combines cell-by-cell AMR with the sub-cell limiter by extending AMR operations to the limiter’s sub-grid representation. Projection and averaging transfer that representation between refinement levels.

5.1. Summary of the cell-by-cell AMR implementation

  • The cell-by-cell AMR grid uses active cells, virtual children, and virtual mothers, with neighboring cells restricted to the same or adjacent refinement levels.Active cells are updated by ADER-DG; virtual children use L2 projection from a mother’s polynomial, while virtual mothers average their children recursively.Source: §5.1 · pp. 8–9
  • Refinement and recoarsening use χ thresholds based on second derivatives of an indicator Φ; Euler tests use density, and discontinuities receive nearby extra refined cells.The typical χref and χrec ranges are approximately [0.2, 0.25] and [0.05, 0.15], respectively; usually one or two neighboring cells are additionally refined.Source: §5.1 · pp. 8–9

5.2. Incorporation of the sub-cell limiter into the AMR framework

  • Limiter status propagates between mothers and children, troubled cells cannot be recoarsened, and AMR projection and averaging must also handle the limiter’s sub-grid representation.Virtual children inherit their active mother’s status, and a troubled active child also flags its virtual mother.Source: §5.2 · p. 9

5.2.1. DG limiter - AMR projection

  • For coarse-to-fine projection in troubled cells, WENO reconstructs sub-grid data because it is designed to remain essentially non-oscillatory around discontinuities while retaining high-order accuracy.The finer-level sub-cell averages are computed from the coarse-level sub-grid representation.Source: §5.2.1 · p. 10

5.2.2. DG limiter - AMR averaging

  • For fine-to-coarse transfer, known sub-cell averages are averaged dimension by dimension, so no reconstruction is required.Source: §5.2.2 · p. 10

6.1. Euler equations of compressible gas dynamics

This section tests the ADER-DG-AMR scheme on classical Euler wave and flow problems, spanning smooth convergence tests and discontinuous-flow benchmarks.

6.1. Euler equations of compressible gas dynamics

  • The Euler tests cover classical wave propagation; blue marks successfully evolved unlimited cells, while red marks troubled cells requiring the sub-cell limiter.The section introduces the conserved variables and fluxes for the Euler system.Source: §6.1 · pp. 10, 12

6.1.1. Numerical convergence study

  • Computed convergence orders agree well with nominal orders through N=8, confirming high-order accuracy with AMR and time-accurate local time stepping.The isentropic-vortex study used one refinement level; for N=8 it used refinement factor r=2.Source: §6.1.1 · p. 12, Table 1 · p. 13

6.1.2. Riemann problems

  • Sod and Lax tests resolve contacts within one finest-level cell and sharply capture limited shocks; numerical profiles agree excellently with exact solutions.In a representative P2 run with about 15% limited cells, the limiter's CPU-time overhead was approximately 1.5× the unlimited DG scheme.Source: §6.1.2 · p. 14, Figure 2 · p. 15, Figure 3 · p. 16

6.1.3. Double Mach reflection problem

  • Double Mach reflection simulations resolve shocks, shear waves, vortices, and sound waves; higher DG orders produce richer vortex roll-up.The increase in vortex-type structures is especially evident from P2 to P5.Source: §6.1.3 · p. 17

6.1.4. Forward facing step

  • The forward-facing-step simulation shows good resolution of the physical instability, with AMR and the sub-cell limiter acting where needed.The P5 test used a coarsest grid of 150 × 50 cells and refinement parameters ℓmax=2 and r=4.Source: §6.1.4 · p. 17, Figure 7 · p. 21

6.1.5. 2D Riemann problems

  • In 2D Riemann problems, small-scale features attributed to Kelvin–Helmholtz instability appear in RP2 and RP4; AMR confines limiting to strong discontinuities.The limited discontinuities are resolved within a few cells at the maximum refinement level.Source: §6.1.5 · p. 21, Figure 8 · p. 22

6.1.6. Cylindrical and spherical explosion problem

  • Cylindrical-explosion AMR and uniform-fine-grid solutions are essentially identical and match the reference; the spherical 3D result also compares well.For the spherical case, limiting activates at the shock front, contact discontinuity, and rarefaction-wave head.Source: §6.1.6 · pp. 23–24, Figure 9 · p. 24

6.2. Equations of ideal magnetohydrodynamics

The ideal MHD section introduces divergence cleaning and applies the ADER-DG-P5 scheme with an a posteriori ADER-WENO3 sub-cell limiter to two standard test problems. In both, the AMR results agree well with uniform-grid calculations and published reference solutions.

6.2. Equations of ideal magnetohydrodynamics

  • The MHD system augments ideal-gas equations with GLM divergence cleaning, which transports magnetic divergence errors out of the domain at a prescribed cleaning velocity.The augmented state and flux are used with ADER-DG-P5, an ADER-WENO3 sub-cell limiter, and the Rusanov solver.Source: §6.2 · pp. 24, 27

6.2.1. MHD rotor problem

  • The rotor test models a rapidly rotating dense fluid embedded in resting low-density fluid, generating torsional Alfvén waves that diminish its angular momentum.The AMR computation is compared with an 800 × 800 finest-level uniform grid.Source: §6.2.1 · p. 27
  • The rotor computation agrees excellently with its finest-level uniform-grid solution and published results, while divergence cleaning prevents spurious density and magnetic-pressure oscillations.The limiter activates in troubled zones; the figure shows the AMR grid and troubled cells.Source: §6.2.1 · p. 27, Figure 12 · p. 28, Figure 13 · p. 29

6.2.2. Orszag-Tang vortex system

  • The Orszag–Tang AMR solution agrees very well with the uniform-grid reference and published fifth-order WENO, unstructured third-order WENO, and ADER-WENO results.The comparison uses times t = 0.5, 2.0, 3.0, and 5.0.Source: §6.2.2 · p. 29

7. Conclusion

The conclusion extends a posteriori sub-cell ADER-WENO limiting to space-time AMR in two and three dimensions, coupling sub-cell averages across refinement levels. Tests cover Euler and ideal MHD equations, verify nominal convergence through degree N = 8, and report improved resolution of fluid dynamics.

7. Conclusion

  • The method combines ADER-DG, a posteriori sub-cell ADER-WENO limiting, and cell-by-cell AMR, with projection and averaging extended across refinement levels.Troubled-cell solutions are reconstructed from the previous time level on 2N + 1 sub-cells and recomputed using ADER-WENO evolution.Source: §7 · p. 29
  • Tests for Euler and ideal MHD equations in two and three dimensions verify nominal convergence through polynomial degree N = 8.The conclusion also reports that the approach revealed unexpected details in fluid dynamics.Source: §7 · pp. 29, 31
  • The authors identify stiff source terms and true physical dissipative processes as future extensions of the method.They also propose extending the adaptive algorithm to high-order semi-implicit DG schemes.Source: §7 · p. 31
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