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Efficient, High Accuracy ADER-WENO Schemes for Hydrodynamics and Divergence-Free Magnetohydrodynamics

Dinshaw S. Balsara, Tobias Rumpf, Michael Dumbser, Claus-Dieter Munz

arXiv:0811.2200v1physics.comp-ph

TL;DR

Accurate, high-order simulation of hydrodynamical and magnetohydrodynamical flows remains challenging, particularly for methods intended to support adaptive meshes. This paper develops ADER-WENO finite-volume schemes with efficient structured-mesh reconstruction and reports design-accuracy attainment and robust performance on multidimensional Euler and MHD tests.

  • Problem

    Existing finite-volume WENO schemes meet design accuracies but do not readily support non-uniform or hierarchical meshes needed for adaptive mesh refinement.

  • Method

    The paper develops ADER-WENO schemes combining efficient multidimensional finite-volume WENO reconstruction with a high-order ADER formulation for Euler and MHD flows.

  • Results

    The ADER-WENO schemes meet their design accuracies in two and three dimensions for Euler and MHD flows and perform very well across the presented tests.

  • Takeaways & Limitations

    The presented ADER-WENO schemes provide accurate and robust performance across the reported multidimensional Euler and MHD test problems.

  • Takeaways & Limitations

    The discussed schemes do not readily support non-uniform or hierarchical meshes.

Abstract

from arXiv · show

The present paper introduces a class of finite volume schemes of increasing order of accuracy in space and time for hyperbolic systems that are in conservation form. This paper specifically focuses on Euler system that is used for modeling the flow of neutral fluids and the divergence-free, ideal magnetohydrodynamics (MHD) system that is used for large scale modeling of ionized plasmas. Efficient techniques for weighted essentially non-oscillatory (WENO) interpolation have been developed for finite volume reconstruction on structured meshes. We also present a new formulation of the ADER (for Arbitrary Derivative Riemann Problem) schemes that relies on a local continuous space-time Galerkin formulation instead of the usual Cauchy-Kovalewski procedure. The schemes reported here have all been implemented in the RIEMANN framework for computational astrophysics. We demonstrate that the ADER-WENO meet their design accuracies. Several stringent test problems of Euler flows and MHD flows are presented in one, two and three dimensions. Many of our test problems involve near infinite shocks in multiple dimensions and the higher order schemes are shown to perform very robustly and accurately under all conditions.

1 Physics Department, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN, 46556, USA · 2 Laboratory of Applied Mathematics, University of Trento, Via Mesiano 77, I-38100 Trento, Italy · 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany

The paper develops modal-space WENO and ADER-CG formulations for high-order Euler and divergence-free MHD schemes, with implementation-oriented descriptions and matching spatial-temporal accuracy. Implemented in RIEMANN, the one-step ADER-WENO methods achieve design accuracies and robust performance on demanding multidimensional flows.

  • 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany: Modal-space interpolation provides a compact WENO reconstruction with explicit formulas through fourth-order spatial accuracy.The reconstruction is formulated in modal space, with explicit formulas for schemes up to fourth order.
  • 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany: Divergence-free MHD evolution defines magnetic-field components and moments on zone faces, yielding a high-order in-zone field representation.The face-based specification naturally supplies an appropriately high-order representation within each zone.
  • 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany: ADER-CG schemes are formulated in modal space with explicit three-dimensional structured-mesh formulas through fourth order.These schemes temporally evolve the WENO-based spatial reconstruction.
  • 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany: ADER-WENO schemes provide temporal accuracy that matches the spatial accuracy of the underlying WENO reconstruction.The formulation uses a local continuous space-time Galerkin approach rather than the usual Cauchy-Kovalewski procedure.
  • 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany: The methods were implemented in the RIEMANN framework for computational astrophysics and use a one-step update.Their one-step structure makes them low-storage alternatives to Runge-Kutta time discretization and suitable building blocks for adaptive mesh refinement.
  • 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany: The ADER-WENO schemes meet their design accuracies in stringent one-, two-, and three-dimensional Euler and MHD tests.The test suite includes multidimensional problems involving near-infinite shocks.
  • 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany: Higher-order schemes perform robustly and accurately under all tested conditions, including multidimensional near-infinite shocks.The increased computational complexity with order is reported to be offset by increased accuracy.
  • 3 Institut für Aerodynamik und Gasdynamik, University of Stuttgart, Pfaffenwaldring 21, D-70550, Stuttgart, Germany: ADER-WENO schemes are presented as worthy alternatives to standard second-order schemes for compressible Euler and MHD flows.This conclusion follows from their accuracy, robustness, and computational tradeoff across the reported tests.

1) Introduction

The paper develops finite-volume ADER-WENO schemes beyond second-order accuracy on structured meshes, combining matched spatial-temporal accuracy with one-step updates suitable for AMR. It extends this approach to divergence-free MHD and implements the schemes in the RIEMANN code for astrophysical fluid dynamics.

  • Motivation: Finite-difference WENO schemes meet design accuracy but do not handle non-uniform or hierarchical meshes well, motivating finite-volume WENO schemes for AMR.Finite-volume schemes can be used as building blocks for adaptive mesh refinement calculations.
  • Hydrodynamics: The paper catalogs finite-volume WENO schemes that go beyond second-order accuracy on structured meshes.This is the paper’s primary stated purpose for hydrodynamical calculations.
  • ADER-WENO formulation: The proposed finite-volume ADER-WENO schemes provide a one-step temporal update with spatial accuracy matched by corresponding temporal accuracy.This structure is intended to support higher-order AMR work while retaining balanced performance.
  • Divergence-free MHD: For divergence-free MHD, the paper presents modern ADER-WENO schemes with one-step updates and spatial and temporal accuracy beyond second order.The schemes are intended as building blocks for AMR-MHD.
  • Implementation: The schemes are implemented in the RIEMANN code for astrophysical fluid dynamics and have been successfully applied to numerous astrophysical applications.The implementation covers the schemes discussed for hydrodynamics and MHD.

2) Efficient, Multi-Dimensional WENO Reconstruction on Structured Meshes · 2.1) One-Dimensional WENO Formulation

The paper develops efficient finite-volume WENO reconstruction on structured meshes by simplifying modal, dimension-by-dimension formulations and reducing the cost of cross-term recovery. The one-dimensional formulation provides compact reconstructions for third-, fourth-, and fifth-order schemes, with stability emphasized in the third-order weighting.

  • 2) Efficient, Multi-Dimensional WENO Reconstruction on Structured Meshes: High-order schemes first reconstruct flow variables in every computational zone to the desired accuracy.This reconstruction is the initial step in the scheme design.
  • 2) Efficient, Multi-Dimensional WENO Reconstruction on Structured Meshes: Tensor products of orthogonal Legendre polynomials provide a modal multidimensional basis with a diagonal mass matrix.The variable is reconstructed from polynomial coefficients, including coordinate-direction and cross terms.
  • 2) Efficient, Multi-Dimensional WENO Reconstruction on Structured Meshes: Finite-volume reconstruction must recover cross terms, unlike pointwise WENO, and structured meshes permit dimension-by-dimension evaluation of coordinate-direction modes.The paper presents a modal simplification of the Jiang–Shu and Balsara–Shu expressions.
  • 2) Efficient, Multi-Dimensional WENO Reconstruction on Structured Meshes: The paper introduces an even less expensive strategy for obtaining the remaining cross terms in multidimensional reconstruction.This advance is catalogued in Sub-Section 2.2, while divergence-free magnetic-field reconstruction is catalogued in Sub-Section 2.3.
  • 2.1) One-Dimensional WENO Formulation: The one-dimensional modal formulation limits non-cross-term modes along each dimension and yields expressions more compact than earlier formulations.It can be applied along each coordinate direction in the multidimensional reconstruction.
  • 2.1.1) Third Order Reconstruction in One Dimension: Third-order reconstruction uses three left-biased, central, and right-biased stencils built from neighboring zone-averaged variables.The reconstructed polynomial contains the zone average and first- and second-order modal terms.
  • 2.1.1) Third Order Reconstruction in One Dimension: 100 times larger is the central stencil S2 linear weight than the weights of one-sided stencils S1 and S3.The choice prioritizes reconstruction stability over an increase in accuracy, and smoothness measures are raised to the fourth power for nonlinear weights.
  • 2.1.2) Fourth Order Reconstruction in One Dimension: Fourth- and fifth-order one-dimensional reconstructions use four and five stencils, respectively, with polynomial bases extended through third- and fourth-order modal terms.The fourth-order smoothness measure is explicitly presented as positive; the fifth-order reconstruction is included as a building block although no fifth-order scheme is presented.

2.2) WENO Formulation for the Cross-Terms

The WENO formulation efficiently reconstructs cross-terms by combining dimension-by-dimension information with smaller stencils, supporting structured-mesh schemes through fourth order. Third- and fourth-order reconstructions use multiple candidate stencils and smoothness measures to obtain non-linearly weighted cross-term values.

  • 2.2) WENO Formulation for the Cross-Terms: Most terms are evaluated by dimension-by-dimension limiting, while remaining cross-terms are gathered efficiently with smaller stencils on structured meshes.This strategy is efficient through fourth order, although it is not generally extensible to all orders.
  • 2.2.1) Third Order Reconstruction of the Cross-Terms: Third-order reconstruction obtains all non-cross modes and the zone average through dimension-by-dimension reconstruction, leaving only the cross-term to evaluate.The known moments belong to the zone labeled “0,0”.
  • 2.2.1) Third Order Reconstruction of the Cross-Terms: Third-order cross-term reconstruction uses four diagonal-zone evaluations, each combining known moments in the zone of interest with one neighboring zone average.The four evaluations provide candidate values for the cross-term.
  • 2.2.1) Third Order Reconstruction of the Cross-Terms: The third-order smoothness measure is IS = 4u_xx^2 + 4u_yy^2 + u_xy^2, and it supplies the non-linearly weighted cross-term value.The same formulae extend the reconstruction to the yz- and xz-planes, completing third-order WENO interpolation on structured meshes.
  • 2.2.2) Fourth Order Reconstruction of the Cross-Terms: Fourth-order reconstruction determines non-cross modes dimension by dimension and evaluates cross-terms using five stencils, including a central stencil added for stability.The central stencil’s linear weight is hundred times larger than those of the four directionally-biased stencils.
  • 2.2.2) Fourth Order Reconstruction of the Cross-Terms: The five fourth-order stencils provide five possible evaluations of the cross-terms, with the central stencil assigned a linear weight 100 times larger than the others.The four alternatives are directionally biased, whereas the central stencil is included for stability.
  • 2.2.2) Fourth Order Reconstruction of the Cross-Terms: Fourth-order smoothness measures are formed by integrating squared second and third derivatives over the zone, either separately or as a combined measure.The reported measures are IS=3(u_xxx^2+u_yyy^2)+4(u_xx^2+u_yy^2)+u_xy^2+3(u_xxy^2+u_xyy^2) and IS=36(u_xxx^2+u_yyy^2)+4(u_xxy^2+u_xyy^2).
  • 2.2.2) Fourth Order Reconstruction of the Cross-Terms: The remaining fourth-order cross-term uses eight spatial diagonal stencils, whose smoothness measures yield a non-linearly weighted value and complete fourth-order WENO interpolation.The eight stencil values use diagonal zones surrounding the element indexed by “0,0,0”.

2.3) Cataloguing Divergence-Free Reconstruction of the Magnetic Field

The divergence-free MHD reconstruction obtains face-centered magnetic-field moments through WENO techniques and uses them with the divergence-free condition to reconstruct the field throughout each zone. Order-specific subsets of face terms provide the required accuracy, while additional divergence-free coefficients act as non-evolutionary terms in the ADER update.

  • Face and interior reconstruction: Face-centered magnetic-field moments are reconstructed with techniques from the preceding subsections, then limited in the corresponding coordinate planes using WENO strategies.The x-, y-, and z-component moments are obtained by limiting in the yz-, xz-, and xy-planes, respectively.
  • Order-specific reconstruction: Second-, third-, and fourth-order schemes use progressively larger subsets of the face-expansion terms: the first line, first two lines, or all three lines.The arrows identify the minimum subset needed for each desired accuracy order.
  • Face and interior reconstruction: The divergence-free reconstruction fully specifies all moments required for third-order accuracy inside the unit cube, and this observation extends to all orders.Face moments combined with the divergence-free condition determine the interior reconstruction coefficients.
  • ADER update: Coefficients present because of the divergence-free condition but unnecessary for third-order accuracy do not participate in ADER time evolution, yet enter flux evaluation and boundary Riemann solvers.These coefficients provide non-evolutionary terms in the ADER update and must be included wherever possible.

3) ADER-CG Formulation

The ADER-CG formulation replaces the Cauchy-Kovalewski procedure with a local continuous Galerkin space-time approach based on pointwise flux evaluations and iterative convergence. It provides an efficient modal formulation for structured meshes, while sacrificing the ability to handle stiff source terms; third-order implementation details are also given.

  • 3) ADER-CG Formulation: ADER-CG uses a local weak space-time formulation requiring only flux evaluations at point values and iterative convergence to the local space-time solution.The formulation is presented as a continuous Galerkin representation in space and time.
  • 3.1) General Formulation of ADER-CG Schemes for Structured Meshes: The general scheme evolves nonlinear hyperbolic conservation laws with conserved variables, fluxes, and a non-stiff source term on a mapped space-time reference element.Each zone is mapped to a unit cube, with the fourth reference coordinate representing time.
  • 3.1) General Formulation of ADER-CG Schemes for Structured Meshes: Flux modes associated with the spatial-only basis functions are evaluated once at τ=0, producing substantial computational-complexity savings.This simplification is the basis of the method’s efficiency and reduces the number of flux evaluations.
  • 3.1) General Formulation of ADER-CG Schemes for Structured Meshes: The ADER-CG simplification relinquishes a weak time formulation and therefore cannot handle stiff source terms, unlike ADER-DG schemes.The stated efficiency advantage is accompanied by this limitation.
  • 3.1) General Formulation of ADER-CG Schemes for Structured Meshes: ADER-CG represents the solution, fluxes, and source terms in modal basis functions whose first LS modes are spatial-only and remaining modes carry time evolution.The same modal construction is applied to the flux components and source terms.
  • 3.1) General Formulation of ADER-CG Schemes for Structured Meshes: The Galerkin projection yields a linear algebraic system involving mass, time-stiffness, and flux-stiffness matrices for the modal coefficients.Space-time integration over the reference element defines the matrix system.
  • 3.1) General Formulation of ADER-CG Schemes for Structured Meshes: “M” iterations of the ADER-CG system are reported to achieve the requisite accuracy of an Mth-order scheme, including in stringent test problems.The iterations are described as inexpensive, and the tests used the minimum requisite number.

4) Flux Calculation , Time-Update and a Step-by-step Description of the ADER-

This section describes the ADER-WENO one-step update, including space-time flux averaging and divergence-free magnetic-field evolution through edge-centered electric fields. It also outlines the reconstruction workflow and reports computational advantages over Runge-Kutta time updates.

  • 4.1 One-step update and flux calculation: High-order updates require appropriately accurate space-time averages of fluxes at zone faces.The ADER space-time representation supplies the quantities needed to average fluxes over each face.
  • 4.1 One-step update and flux calculation: Divergence-free magnetic evolution updates face-centered magnetic fields using edge-centered, space-time-averaged electric fields derived from upwinded fluxes.At each edge, contributions from four adjoining faces are combined by arithmetic averaging.
  • 4.1 One-step update and flux calculation: For HLL fluxes, the third square bracket is doubled only for electric-field evaluation when 0 < λ_L < λ_R.In all other cases, the HLL flux expression is used directly for the electric fields.
  • 4.2 Step-by-step ADER-WENO scheme: The ADER-WENO time step first reconstructs face-centered magnetic-field moments without characteristic interpolation, then reconstructs zone-centered quantities in characteristic space.Magnetic-field reconstruction also provides a zone-centered mean magnetic field with the requisite accuracy, while cross-term reconstruction is included for zone-centered quantities.
  • 4.2 Step-by-step ADER-WENO scheme: ~31,000 zones per second are reconstructed and updated in three dimensions on a single core.The paper further states that higher-order schemes’ added computational complexity is offset by increased accuracy and is considerably less expensive than Runge-Kutta time updates.

5) Order Property

The second-, third-, and fourth-order ADER-WENO schemes meet their expected accuracy orders in one-, two-, and three-dimensional tests. Higher-order schemes achieve comparable or greater accuracy on substantially coarser meshes, although the fourth-order vortex result uses a larger domain.

  • Unmagnetized vortex: All three schemes meet the expected order of accuracy even for a small number of zones in the unmagnetized vortex test.Errors were measured using the density variable.
  • Unmagnetized vortex: The third order scheme at 128x128 zones achieves an L1 error norm comparable to the second order scheme at 256x256 zones.This demonstrates the advantage of the higher-order scheme.
  • Unmagnetized vortex: The fourth order scheme also meets its expected accuracy order, but cannot be directly compared because it uses a much larger computational domain.The fourth-order calculation uses [-10, 10] x [-10, 10], whereas the smaller-domain tests use [-5, 5] x [-5, 5].
  • Magnetized vortex: The magnetized vortex tests show all three schemes meeting the design order of accuracy at a small number of zones.The error was measured in the x-component of the magnetic field.
  • Magnetized vortex: The third order scheme at 128x128 zone resolution matches the second order scheme’s L1 error at 256x256 zone resolution, using half as many zones in each direction.Its slightly greater computational complexity is offset by the comparable accuracy on the coarser mesh.
  • Torsional Alfvén wave: In the torsional Alfvén wave test, all three methods meet the expected order of accuracy even for a small number of zones.The test is conducted at very high plasma-β to suppress instability growth, and the magnetic field is initialized divergence-free.

6) Hydrodynamical Test Problems

The ADER-WENO schemes accurately and robustly solve stringent hydrodynamical test problems, resolving shocks, extrema, and multidimensional flow structures with relatively few zones. Higher-order schemes achieve solution quality comparable to lower-order schemes at substantially smaller resolutions.

  • Section overview: The schemes operate accurately and robustly across the presented hydrodynamical test problems.All tests were implemented in the RIEMANN code.
  • Interacting blast problem: Fourth-order ADER-WENO captures the interacting-blast flow structures closely to the converged solution using 400 zones.The left-going contact discontinuity is captured well, and all 400-zone structures are very close to the 1600-zone converged simulation.
  • Shock-entropy wave interaction: Fourth-order ADER-WENO nearly converges to the shock-entropy-wave reference solution on a 200-zone grid and preserves all reference extrema.It needs no more than 11 points between density extrema immediately after the shock, while the third-order scheme remains close to the reference.
  • Shock-entropy wave interaction: The ADER-WENO schemes converge to the shock-entropy-wave reference solution with fewer points and smaller fixed-zone error than lower-order schemes.The TVD scheme misses the reference solution by a wide margin, whereas the third-order ADER-WENO scheme converges faster and has smaller error at constant zone count.
  • Forward-facing step: In the forward-facing-step problem, fourth-order ADER-WENO sharply captures shocks and resolves the vortex sheet with only a few zones across it.At 960x320 and 480x160 resolutions, vortex-sheet roll-up is clearly visible with little or no spreading.
  • Double Mach reflection: Fourth-order ADER-WENO resolves the double-Mach-reflection structures more efficiently than second-order schemes.A second-order scheme would require at least four times as many zones in each direction and more CPU time to resolve the instability.

7) MHD Test Problems

The MHD test problems show that ADER-WENO schemes operate accurately and robustly across stringent one-, two-, and multidimensional cases implemented in RIEMANN. They sharply resolve discontinuities and strong shocks, while higher order and divergence-free formulations reduce dissipation and preserve positive pressure in low-β plasmas.

  • Riemann problems: The fourth-order ADER-WENO scheme captures all Riemann-problem shock profiles within a few zones, including slow shocks, contacts, and rotational discontinuities.Resolving every discontinuity sharply distinguishes rotational discontinuities from nearby slow shocks and maintains high accuracy.
  • Riemann problems: For colliding magnetized streams producing Mach 25.5 fast shocks, shock profiles are captured within a few zones without post-shock oscillations.The test demonstrates high resolution, high accuracy, and low numerical dissipation for very strong shocks.
  • Alfvén-wave decay: Increasing ADER-WENO order significantly reduces numerical dissipation independently of the Riemann solver, making higher order schemes favorable for wave-propagation simulations.The linearized solver is less dissipative than HLL, but this difference decreases at higher order as reconstruction improves.
  • Rotor problem: The multidimensional limiting performs well in the rotor problem, consistently reproducing the established physical descriptions.This result supports the effectiveness of the paper’s multidimensional limiting procedure.
  • Low-β plasma tests: In low-β plasma tests, the fourth-order divergence-free reconstruction crisply captures nearly infinite magnetosonic shocks without undue oscillations and maintains positive pressure.The results identify divergence-free reconstruction as a significant improvement for low-β plasma simulations.

8) Conclusions

The paper presents general high-order ADER-WENO schemes for hyperbolic conservation laws, successfully applied to Euler and divergence-free MHD flows. Its conclusions emphasize efficient reconstruction, compact one-step ADER-CG updates, robust multidimensional accuracy, AMR suitability, and accuracy gains that offset increased computational complexity.

  • Overall contributions: ADER-WENO schemes provide a general high-order framework for hyperbolic conservation laws and have been successfully applied to Euler and MHD flows.The methods are described as general and applicable to several hyperbolic systems.
  • Reconstruction: Modal-space finite-volume WENO reconstruction on structured meshes is efficient, compact, and explicitly developed through fourth-order spatial accuracy.The formulation is presented as especially elegant and compact when interpolating functions are expressed in modal space.
  • Reconstruction: Divergence-free magnetic-field reconstruction supplies all magnetic-field moments within a zone consistently with a specified retained order.The paper also discusses the essential aspects of divergence-free reconstruction and refers to further implementation details elsewhere.
  • ADER-CG formulation: Modal-space ADER-CG schemes yield compact formulations for structured meshes, while the framework also supports structured and unstructured meshes.Dubiner bases provide a similarly compact and elegant formulation on unstructured meshes.
  • Efficiency and AMR: One-step ADER-WENO updates reduce storage relative to multistage Runge-Kutta discretizations, avoid repeated stage reconstructions, bypass Butcher barriers, and suit AMR calculations.The conclusions identify one-step updates as more efficient alternatives and desirable building blocks for AMR.
  • Validation and accuracy: Tests show design accuracies in two and three dimensions and very good performance on demanding Euler and MHD problems involving almost infinite shocks.The conclusions state that increased-order computational complexity is offset by increased accuracy, making ADER-WENO schemes worthy alternatives to standard second-order schemes.

Appendix A) Flattening Algorithm in the Vicinity of Strong Shocks

The appendix presents a strong-shock detector based on velocity divergence and local minimum magnetosonic speed, then uses it to flatten higher-order reconstructions while preserving linear structure. It also blends linearized Riemann and HLL fluxes to address strong shocks and rarefactions.

  • Appendix A) Flattening Algorithm in the Vicinity of Strong Shocks: The detector compares each zone’s undivided velocity divergence with a local minimum magnetosonic speed scanned over nine zones in two dimensions or twenty-seven in three dimensions.It is designed to activate smoothly near strongly compressive shocks while remaining zero in smooth flow and moderately compressive shocks.
  • Appendix A) Flattening Algorithm in the Vicinity of Strong Shocks: The detector function is zero in smooth flow and moderate shocks, but rises smoothly toward unity near strongly compressive shocks to retain the scheme’s order property.A modified detector can also handle strong rarefactions when they become problematic.
  • Appendix A) Flattening Algorithm in the Vicinity of Strong Shocks: In detected strong-shock zones, quadratic and cubic moments are effectively zeroed, while linear variations are blended with a slope limiter to preserve some structure.The first-moment treatment anticipates shocks about to cross a zone and blends WENO slopes with MinMod-limited slopes using a factor χ.
  • Appendix A) Flattening Algorithm in the Vicinity of Strong Shocks: The detector blends a linearized Riemann solver with an HLL flux, yielding pure HLL fluxes for strong shocks or rarefactions and pure linearized fluxes for weak ones.Intermediate situations receive a linear blend, addressing the linearized solver’s carbuncle instability and poor behavior in strong rarefactions.

Accuracy

The Accuracy section specifies second- and fourth-order ADER-CG implementations, including their update iterations and nodal-to-modal flux transcription procedures. It describes symmetric reference-element nodes for constructing these formulations.

  • Second-order ADER-CG: Second-order ADER-CG is formulated through a resultant iteration and nodal-to-modal transcription of fluxes at τ=0 and τ>0.The transcription uses symmetrically placed nodes in the reference element.
  • Fourth-order ADER-CG: Fourth-order ADER-CG likewise defines a resultant iteration and separate nodal-to-modal flux transcriptions at τ=0 and τ>0.Its construction uses symmetrically placed nodes in the reference element.
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