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Fully Discrete Active Flux Methods for the Euler Equations: Exploring Different Reconstructions and New Multi-Dimensional Evolution Operators
Christiane Helzel, Mária Lukáčová-Medvid'ová, Amelie Porfetye, Zhuyan Tang
TL;DR
Fully discrete Active Flux methods for the two-dimensional Euler equations depend on approximations that evolve point values, and the stability and accuracy effects of those choices require examination. The paper derives new third-order bicharacteristic evolution operators and compares them with different reconstructions. CWENO permits time steps up to CFL≤0.7, while continuous AF can be more accurate on coarse grids but less robust.
Problem
Fully discrete Active Flux methods need point-value evolution operators, and the accuracy and stability effects of approximate operators and reconstruction choices remain to be examined.
Source: §1 · pp. 1–2Method
The authors derive third-order multidimensional bicharacteristic evolution operators for locally linearized Euler equations and compare continuous AF and CWENO reconstructions.
Source: §1 · p. 2Results
CWENO permits time steps up to CFL≤0.7; both methods show third-order convergence on smooth tests, with smaller errors for continuous AF, which can be less robust.
Source: Conclusions · p. 35Takeaways & Limitations
The reconstruction and evolution operator involve a stability–accuracy trade-off: the more local continuous-AF stencil can be more accurate, particularly on coarse grids, while lower numerical viscosity makes it less robust.
Source: Conclusions · p. 35The reported rankings reflect combined spatial and temporal discretization effects and do not isolate reconstruction or evolution-operator effects.
Source: §6.5 · p. 29
Abstract
from arXiv · showhide
We present new fully discrete Cartesian grid Active Flux methods for the two-dimensional Euler equations of gas dynamics. A key component of these methods is the evolution operator used to evolve point-value degrees of freedom. Therefore, we derive several new, third-order accurate, multidimensional evolution operators for point-value updates of the linearized Euler equations using the method of bicharacteristics. We also investigate how the reconstruction and choice of approximate evolution operator affect the accuracy and stability of the resulting methods.
1 Introduction
The paper extends fully discrete Active Flux (AF) methods for the two-dimensional Euler equations with new point-value evolution operators, while comparing continuous AF and CWENO reconstructions. It examines how these choices affect stability, accuracy, and robustness.
1 Introduction
- AF methods evolve boundary point values as independent degrees of freedom alongside cell averages, and earlier acoustic methods showed desirable properties including vorticity preservation.The paper connects this motivation to the hypothesis that controlling vorticity may reduce susceptibility to anomalous Euler solutions.Source: §1 · p. 1
- Method-of-lines AF variants avoid evolution operators for point values, but their time-stepping stages enlarge the stencil and stability requirements appear to impose severe time-step restrictions.Related fully discrete methods for the Euler equations use equation-specific structure, whereas this paper uses bicharacteristics, a more general approach for linear hyperbolic systems.Source: §1 · pp. 2–3
- The paper extends the third-order fully discrete AF method for the two-dimensional Euler equations by incorporating new acoustic evolution operators into its point-value updates.The existing method uses local linearizations and a corrected point-value update.Source: §1 · p. 2
- The study compares the classical globally continuous AF reconstruction with CWENO, examining how reconstruction and evolution-operator choices affect stability, accuracy, and robustness.CWENO uses nonlinear weights to help suppress oscillations near discontinuities, but is not globally continuous.Source: §1 · p. 2
- CWENO reconstruction uses neighboring cell averages and has a wider stencil; in the authors’ acoustic study, it improved linear stability properties.The paper uses the comparison with classical AF to examine whether discontinuous reconstruction is advantageous.Source: §1 · p. 2
2 Governing Equations and Evolution Operators
The section sets up the two-dimensional Euler equations and locally linearizes them for point-value evolution. Bicharacteristics yield exact integral updates, while EG2 approximates computationally intractable mantle integrals; nonlinear point updates require a correction for third-order accuracy.
2.1 Euler Equations and their Local Linearization
- The governing equations are written in conservative form for cell-average updates and quasilinear primitive-variable form for point-value updates based on local linearization.The local Jacobians are frozen at evaluation points corresponding to cell corners and edge midpoints, which also serve in flux integration.Source: §2 · p. 3, §2.1 · p. 4
- Freezing the Jacobians at a reference state produces a constant-coefficient linearized system whose characteristic speeds represent acoustic waves and advected entropy and shear waves.The acoustic eigenvalues are the background-flow speed projected onto the direction plus or minus the local sound speed; the other two modes are advected.Source: §2.1 · p. 4
2.2 Exact Integral Equations via Bicharacteristics for the Linearized Problem
- The method of bicharacteristics gives exact point-value evolution through data on the advected cone base and multidimensional interactions integrated over its mantle.The cone’s base perimeter is parameterized by Q(θ), while P′ is its purely advected center.Source: §2.2 · pp. 4–6
2.3 The EG2 Approximate Evolution Operator
- EG2 approximates the exact mantle integrals with a time-rectangle rule and updates primitive variables using data on the advected base circle and center.Although formally third-order accurate, EG2 has restrictive CFL conditions with AF reconstruction and does not exactly preserve one-dimensional planar waves for piecewise quadratic data.Source: §2.3 · p. 6
2.4 Point Value Approximation of the Nonlinear Equations
- For nonlinear point updates, local linearization alone is generally not third-order accurate; adding a correction that cancels its leading error term restores third-order accuracy.The paper uses a local linearization whose leading second-order error term has a particularly simple structure.Source: §2.4 · p. 7
3 Fully Discrete AF and AFCW Methods
The methods combine quadratic reconstruction, space-time flux quadrature, and evolved point values to form fully discrete AF and AFCW schemes. They differ in which data reconstruct the solution and in the associated point-value evolution and local linearization procedures.
3 Fully Discrete AF and AFCW Methods
- Both fully discrete formulations combine piecewise-quadratic reconstruction, space-time flux quadrature, and point-value updates for third-order accuracy.The section also introduces simplifications used to analyze linear stability.Source: §3 · p. 7
3.1 Approximation of Cell Average Values
- Cell averages are updated by a finite-volume scheme using numerical fluxes approximated by space-time quadrature.The methods use quadrature rules to approximate the flux integrals on cell faces.Source: §3.1 · p. 7
- Classical AF uses Simpson’s rule in space and time, while AFCW uses Simpson’s rule in space with Gauss quadrature in time because its discontinuous reconstruction can provide multiple traces at the initial time.Both rules require evolved point values at temporal quadrature nodes.Source: §3.1 · p. 8
3.2 Reconstruction
- The continuous AF reconstruction uses point values and cell averages, whereas the piecewise-quadratic CWENO reconstruction uses cell averages alone to supply initial data for evolution.For AF, primitive-variable cell averages are computed using transformed point values and Simpson’s rule; CWENO obtains them by approximating the conservative-to-primitive transformation for smooth solutions.Source: §3.2 · pp. 8–9
3.3 Local linearization
- Third-order point-value updates require suitable local linearization states and a correction that discretizes the leading linearization-error term.The chosen states depend on temporal quadrature; successive point values can serve as linearization states for later evolutions under Simpson’s rule.Source: Local linearization for smooth solutions · p. 10
Local linearization for smooth solutions
- For smooth solutions, the local linearization states are constructed to meet the accuracy condition at the temporal quadrature points used by each method.The CWENO procedure constructs states for its two Gauss nodes, while the continuous-reconstruction method uses successive point-value updates.Source: Local linearization for smooth solutions · pp. 10–11
Shock indicator
- A shock indicator detects shock waves and reflects their strength, modifying local linearization states near shocks and contributing to AF limiting.It combines pressure-gradient information with velocity divergence and vorticity; θ is near 1 in smooth regions and decreases toward 0 near shocks.Source: Shock indicator · p. 11
Local linearization near discontinuous solution structures
- Near shocks, the continuous-AF linearization blends the smooth-solution state toward the average of surrounding cell averages, trading smooth-region third-order accuracy for robustness when used alone.The authors combine this shock-region state with the smooth-solution approach; their modified indicator also provides a suitable linearization near transonic shock waves.Source: Local linearization near discontinuous… · pp. 11–12
4 New Evolution Operators for the Linearized Euler Equations
The section develops third-order evolution operators for linearized Euler point values, targeting exactness for quadratic data and improved stability over EG2. It derives EGquad and parameterized EG2 variants, then compares their computational costs and linear stability limits.
4 New Evolution Operators for the Linearized Euler Equations
- EGquad is exact for quadratic plane waves, but with globally continuous AF reconstruction its stability limit remains as restrictive as EG2’s.The section introduces EGquad for use with CWENO and develops further operators to improve stability with globally continuous reconstruction.Source: §4 · p. 12
4.1.1 Exact 1D Solution for Quadratic Data
- The quadratic-data reference states are obtained by evaluating exact one-dimensional solutions at the advected origin in a stationary local frame.The analysis considers a planar wave aligned with the x-axis and quadratic initial profiles for the primitive variables.Source: §4.1.1 · pp. 12–13
4.1.2 Adjusting the EG2 Operator to Capture the Exact Solution
- EGquad replaces EG2’s continuous trigonometric integrals with discrete finite differences to reproduce the exact quadratic-data evolution.EG2’s time-rectangle rule misses the required quadratic coefficient; the replacement yields the approximate operator used with the AFCW method.Source: §4.1.2 · pp. 13–14
4.2 New Evolution Operator with Increased Stability
- EG2δ changes pressure and density updates while retaining the original EG2 velocity formulas; EG2δ,ν additionally modifies velocity updates, with δ·CFL ≤ 0.5 and ν·CFL ≤ 0.5.EG2δ reduces to EG2 when δ = 0.Source: §4.2 · pp. 15–16
4.3 Computational Costs
- The EG2δ and EG2δ,ν updates generally require six additional integrals per point value, but δ = ν = 1 lets three averages reuse EG2 calculations.EGquad has computational costs comparable to EG2, so the cost comparison focuses on the parameterized EG2 operators.Source: §4.3 · p. 16
4.4 Investigation of linear stability
- CFL = 0.444 is the largest reported limit, attained by AFEG2ω with δ = ν = 1 and ω = 0.4, while requiring only three additional integrals.AFEG2δ,ν with δ = 1.0 and ν = 0.4 reaches CFL ≤ 0.440; EG2 and EGquad are limited to CFL = 0.279.Source: §4.4 · pp. 17–18, Figure 3 · p. 17
5 Approximation of Transonic Rarefaction Waves
With continuous AF reconstruction, AFEG2 fails to preserve the Shu vortex in flow regimes related to transonic rarefaction waves. Enlarging the evolution operator’s integration radii avoids the observed artifacts, while AFCW remains stable on the tested grids.
5 Approximation of Transonic Rarefaction Waves
- The AFEG2 failure occurs for certain background speeds in regimes related to transonic rarefaction waves, with |u′| < c′, |v′| < c′, and √(u′² + v′²) > c′.The reported problem is inaccurate approximation of the Shu vortex, rather than an unphysical rarefaction-wave approximation.Source: §5 · p. 19
- This Shu-vortex pathology differs from previously reported AF decoupling, which was associated with characteristic-speed sign changes across shocks rather than transonic rarefaction waves.Changing the local linearization state does not avoid the behavior reported here.Source: §5 · p. 19
- AFEG2 loses the Shu vortex and develops spurious artifacts, whereas AFEG22.0 (δ=2) preserves it without visible artifacts and retains third-order convergence.The comparison is shown at t = 60; the third-order convergence is reported for AFEG22.0.Source: §5 · pp. 19–20, Figure 4 · p. 20
- Enlarging the evolution operator’s integration radii addresses the suspected lack of domain-of-dependence information; δ ≥ 1.8 suffices, and δ=2 is chosen for efficiency.The authors report that the entropy-fix does not reduce the method’s order of accuracy.Source: §5 · p. 20
- AFCW remains stable through t = 60 without limiting or an entropy-fix, and its vortex stays localized without the nonlocal artifacts seen with AFEG2.The authors attribute sufficient stabilization for this Shu-vortex problem to AFCW’s numerical viscosity.Source: §5 · p. 21, Figure 6 · p. 21
- Between 256² and 512² grids, density EOC is 2.83, while total-energy and momentum EOCs range from 2.55 to 2.65.The density is approaching its third-order asymptotic regime; the other components have not yet fully reached it on the grids considered.Source: Figure 5 · p. 20, §5 · p. 21
6 Further Numerical Results
The tests examine accuracy on smooth flows and behavior on discontinuous and instability problems. Both methods show third-order convergence on smooth tests, while their errors, sharpness, and diagnostic rankings vary by reconstruction and test.
6.1 Investigation of Accuracy
- AFEG2.0 shows expected third-order convergence for density advection; AFCW approaches that order but has errors approximately one order of magnitude larger.The comparison covers w0 = 0.5 at t = 4 and w0 = 1.0 at t = 2.Source: §6.1 · p. 22, Figure 7 · p. 23
- EOC 2.96 for AFCW density on the 256²-to-512² refinement step, while its momentum and total-energy EOCs of 2.68–2.69 remain pre-asymptotic; both AF methods show third-order convergence.AFCW errors are more than an order of magnitude larger than those from AFEG2 and AFEG20.4 1.0,1.0.Source: §6.1 · p. 24
6.2 Two-dimensional Riemann problems
- The three methods resolve the tested two-dimensional Riemann structures well; continuous-reconstruction AF methods give slightly sharper Mach-stem contact lines and more pronounced contact-line roll-up.For Configuration 3, all three reproduce the complex wave pattern well; the comparison uses grids from 256² to 1024².Source: §6.2 · p. 25, Figure 9 · p. 26, Figure 10 · p. 30
6.3 Sod shock tube
- The three methods produce qualitatively similar radial profiles for the circular Sod shock tube at t = 0.1.The computation uses bound-preserving and shock-indicating limiting, without flux limiting.Source: §6.3 · p. 27, Figure 12 · p. 32
6.4 Mach 80 Jet
- The Mach 80 jet is tested as an extreme robustness case with point-value and flux limiting; the displayed density and pressure fields compare AFEG2, AFEG2.0, and AFCW at CFL ≤ 0.23.The solutions are shown at t = 0.07 on a 512×256 grid.Source: §6.4 · p. 27, Figure 13 · p. 32
6.5 Kelvin-Helmholtz instability
- For the Kelvin–Helmholtz test, all methods reveal finer vortical structures under refinement, while differences become more pronounced by T = 2; entropy and energy-defect diagnostics distinguish the methods despite nearly identical averaged energies.AFCW has the largest time-averaged entropy and smallest energy defect; entropy and Bregman-distance criteria favor different methods, and these rankings are not overall accuracy rankings.Source: §6.5 · p. 29, Table 2 · p. 29
Conclusions
The paper develops fully discrete, third-order Active Flux and Evolution Galerkin methods for the two-dimensional Euler equations, with new multidimensional point-value evolution operators. It reports improved stability for continuous reconstruction and contrasts that reconstruction’s accuracy with CWENO’s robustness.
Conclusions
- New third-order evolution operators improve the classical continuous-reconstruction AF stability limit from CFL ≤ 0.279 to CFL ≤ 0.444.The operators update point values using approximations to the two-dimensional linearized Euler equations via bicharacteristics.Source: Conclusions · p. 35
- CWENO enlarges the stencil and permits time steps up to CFL ≤ 0.7; both reconstructions achieve third-order convergence on smooth tests, but continuous AF has smaller errors and sharper contact waves.AFCW needed point-value limiting only for the Sod tube, Riemann Configuration 3, and Mach 80 jet; other tests needed no limiting.Source: Conclusions · p. 35
- For the Shu vortex, AFCW accurately approximates the flow, whereas continuous AF with EG2 is unstable; EG2.0 acts as an entropy fix and yields accurate results.Source: Conclusions · p. 35
- Both schemes use approximations to two-dimensional linearized Euler equations, but continuous AF’s lower numerical viscosity is associated with less robustness and greater sensitivity to operator, linearization, and Shu-vortex choices.The more viscous AFCW method was less sensitive in these respects.Source: Conclusions · p. 35