Source-linked AI summary
ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement
Michael Dumbser, Olindo Zanotti, Arturo Hidalgo, Dinshaw S. Balsara
TL;DR
The paper addresses the challenge of combining high-order finite-volume accuracy with efficient space–time AMR and local time stepping. It develops a one-step ADER-WENO method using WENO spatial reconstruction and a local space–time DG predictor, then evaluates it on Euler and MHD problems. The studies confirm high-order convergence, AMR benefits, and a bounded administrative overhead in the tested settings.
Problem
Existing AMR methods commonly pair method-of-lines discretizations with Runge–Kutta time stepping, motivating a one-step high-order alternative for space–time adaptive grids.
Method
The scheme combines WENO polynomial reconstruction, a local space–time discontinuous Galerkin predictor, cell-by-cell tree-based AMR, and MPI parallelization.
Results
Numerical studies confirm high-order accuracy in space and time, benefits of AMR over uniform fine grids, and an administrative overhead of at most 25% in the reported tests.
Takeaways & Limitations
The method supports two- and three-dimensional simulations of Euler and MHD systems while efficiently resolving turbulent structures, vortices, and sound waves.
Takeaways & Limitations
The reconstruction is not optimal because achieving a given accuracy order requires a larger total stencil than the original optimal WENO scheme.
Abstract
from arXiv · showhide
We present the first high order one-step ADER-WENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR property has been implemented 'cell-by-cell', with a standard tree-type algorithm, while the scheme has been parallelized via the Message Passing Interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speed-up with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space--time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.
1 Introduction
The paper combines a high-order one-step ADER-WENO finite-volume method with tree-based, cell-by-cell AMR and time-accurate local time stepping. It addresses the limitations of predominantly Runge–Kutta-based AMR approaches while supporting multidimensional hyperbolic conservation-law simulations.
- Motivation and approach: One-step schemes evolve reconstructed polynomials locally in time to compute higher-order numerical fluxes without multistage Runge–Kutta updates.The approach uses a local space–time Galerkin predictor based on the weak integral form of the PDE.
- Related AMR methods: Most existing high-order AMR methods combine method-of-lines discretizations with higher-order TVD Runge–Kutta time integration.The paper positions its method as an alternative to this common design.
- Contribution: The paper presents the first tree-based cell-by-cell AMR algorithm combined with a finite-volume ADER-WENO one-step update scheme.The one-step formulation directly supports time-accurate local time stepping on space–time adaptive meshes.
- Evaluation: The method is evaluated for the Euler and magnetohydrodynamics equations through multidimensional numerical tests.The paper structure includes an ADER-method description, AMR infrastructure, and numerical experiments for both systems.
2 Numerical Method
The numerical method combines finite-volume evolution, dimension-by-dimension WENO polynomial reconstruction, and a local space–time discontinuous Galerkin predictor. Its reconstructed spatial polynomial is evolved locally to produce the space–time data required for high-order flux evaluation.
- 2.1 The finite volume scheme: The finite-volume scheme integrates balance laws over Cartesian control volumes and uses space–time averaged numerical fluxes and sources.On adaptive meshes, cells may also be addressed with a unique mono-index while the scheme remains written in Cartesian indices.
- 2.1 The finite volume scheme: The method uses Rusanov or Osher–Solomon numerical fluxes to couple neighboring reconstructed states.The Rusanov flux is identified with the local Lax–Friedrichs flux, while the Osher-type flux uses a straight-line path in phase space.
- 2.2 WENO reconstruction: WENO reconstruction builds a data-dependent nonlinear combination of candidate polynomials on dimension-specific stencils.The reconstruction proceeds successively in the x, y, and z directions while preserving integral conservation.
- 2.2 WENO reconstruction: The reconstruction produces complete multidimensional polynomials in a nodal basis rather than only interface point values.This differs from the original optimal WENO formulation and requires a larger total stencil for a given accuracy order.
- 2.3 Local space–time DG predictor: A local space–time Galerkin predictor evolves each reconstructed spatial polynomial into a high-order space–time polynomial.The predictor requires pointwise evaluations of fluxes and source terms rather than algebraic manipulations of space and time derivatives.
3 Adaptive Mesh Refinement
The method uses tree-based, cell-by-cell AMR integrated with one-step ADER-WENO updates on space-time adaptive grids. Refinement maintains local reconstruction and neighbor-level constraints while enabling local time stepping and virtual-cell projection.
- AMR infrastructure: The AMR implementation uses cell-by-cell refinement with a tree-type data structure rather than nested rectangular patches.This choice is described as slightly more general and straightforward to implement.
- AMR infrastructure: Each refined cell generates r^d children, while active, virtual-child, and virtual-mother status flags determine finite-volume updating, projection, or recursive averaging.The coarsest grid is uniform Cartesian, and refinement levels run from ℓ=0 to a prescribed ℓmax.
- AMR infrastructure: The refinement factor must satisfy r≥M so WENO reconstruction remains local on the coarser grid, with virtual ghost cells supplying the required neighborhood.The virtual ghost cells allow the uniform Cartesian reconstruction procedure to be applied on AMR grids.
- AMR infrastructure: Voronoi-neighbor rules constrain refinement: neighboring levels differ by at most one, and virtual children are activated or generated to preserve this condition.A refined cell cannot have a Voronoi neighbor without children when its children have status σ≤0.
- Refinement criterion: The refinement criterion is local and based on a second-derivative error indicator, with thresholds χref≈[0.2,0.25], χrec≈[0.05,0.15], and ε=0.01 in most tests.The filter ε prevents refinement in regions of small ripples.
- Local time stepping: Each refinement level advances with local steps satisfying Δtℓ=rΔtℓ+1, while the one-step ADER-WENO ingredients transfer to AMR with few modifications.The local space-time DG predictor remains element-local even when adjacent cells have different refinement levels.
- Efficiency: The reported CPU-time measurements include AMR overhead from updating, averaging, and projecting virtual ghost cells.The overhead is included in average time per active real-element update.
4 Numerical Tests
The numerical tests apply the ADER-WENO AMR scheme to Euler, MHD, vortex, explosion, instability, and aeroacoustic problems. Results include high-order convergence, close agreement with reference or uniform-grid solutions, and substantial AMR efficiency gains.
- Euler equations: The isentropic-vortex convergence study evaluates third- and fourth-order ADER-WENO versions using density errors at final time t_f = 10.The study compares AMR and uniform-grid configurations.
- Euler equations: Two-dimensional explosion simulations agree very well with the one-dimensional reference solution and differ little from the equivalent uniform fine-grid result.The fourth-order AMR calculation uses a 34 × 34 level-zero grid, refinement factor r = 3, and ℓ_max = 2.
- Euler equations: The three-dimensional explosion test uses 9,079,984 final AMR elements versus an equivalent fine-grid resolution of 28,652,616 cells.A third-order ADER-WENO scheme is run to t = 0.25 from a 34 × 34 × 34 level-zero mesh; the simulation takes 7.5 hours.
- Euler equations: The scheme produces qualitatively good Kelvin–Helmholtz roll-up, which serves as an indicator of the numerical viscosity affecting the shear-wave instability.The results are reported as being in good qualitative agreement with other published results.
- Euler equations: In the co-rotating vortex-pair problem, the fourth-order scheme resolves long-wavelength sound waves while handling strong near-core velocity and pressure gradients.The simulations use about 10 points per wavelength for the fourth-order scheme and continue to t = 500.
- Classical MHD equations: For the MHD test, AMR and uniform fine-mesh solutions agree very well, while divergence cleaning removes spurious density and magnetic-pressure oscillations.The AMR calculation uses a third-order scheme and an equivalent uniform resolution of 800 × 800.
- Euler equations: AMR can reduce computational cost substantially: one fine uniform-mesh computation requires more than five times as many elements and more than seven times the CPU time.This efficiency gain is reported for the corresponding AMR test problem.
5 Conclusions
The paper presents a high-order one-step ADER-WENO finite volume scheme on space-time adaptive AMR grids and validates its accuracy, efficiency, and applicability to multidimensional Euler and MHD problems. The authors also identify extensions toward broader physical models and applications.
- Conclusions: The scheme combines one-step ADER-WENO finite volumes with space-time adaptive AMR grids and time-accurate local time stepping.Its implementation uses a local space-time discontinuous Galerkin predictor and supports cell-by-cell AMR.
- Conclusions: The space-time adaptive mesh administration overhead is at most 25%.This bound is quantified in Table 1.
- Conclusions: Numerical convergence studies confirm that the method reaches its claimed higher order in space and time.The validation covers the proposed discretization rather than a single test configuration.
- Conclusions: Tests in two and three dimensions cover compressible Euler and classical MHD equations, with AMR outperforming uniform fine grids in the reported examples.The MHD rotor comparison includes AMR and a fine uniform grid at the finest AMR level.
- Conclusions: Higher-order AMR improves resolution of small-scale turbulent structures, vortices, and sound waves by requiring little numerical dissipation.The conclusion specifically connects these phenomena with the need for low numerical dissipation.
- Conclusions: Future work targets general PNPM schemes, realistic computational astrophysics, turbulent viscous flows, chemically reacting multiphase flows, and nonconservative hyperbolic systems.These extensions define the current method's stated research direction.