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Multidimensional HLLE Riemann solver; Application to Euler and Magnetohydrodynamic Flows
Dinshaw S. Balsara
TL;DR
The paper develops a multidimensional HLLE Riemann solver and a second-order Euler/MHD scheme based on it. The solver supplies multidimensional dissipation and upwinded electric fields, while the resulting scheme is cost-competitive with one-dimensional-solver schemes and permits larger timesteps.
Problem
Existing multidimensional flux formulations can select upwind terms that differ from those obtained by careful multidimensional analysis, while strong oblique shocks lack y-direction dissipation in the relevant formulation.
Method
The method introduces one constant resolved state between the considered states, derives closed-form resolved fluxes, and combines the multidimensional solver with divergence-free magnetic reconstruction and a predictor-corrector scheme.
Results
The solver provides positively conservative density treatment, and the resulting second-order Euler/MHD scheme is robust, design-accurate, cost-competitive with one-dimensional-solver schemes, and permits larger timesteps.
Takeaways & Limitations
Multidimensional Riemann problems and upwinded electric fields can support an efficient second-order Euler/MHD scheme with larger timesteps than schemes based on one-dimensional Riemann solvers.
Takeaways & Limitations
Strong shocks may require a shock detector to truncate the y-flux contribution because the formulation lacks corresponding y-direction dissipation.
Abstract
from arXiv · showhide
In this work we present a general strategy for constructing multidimensional Riemann solvers with a single intermediate state, with particular attention paid to detailing the two-dimensional Riemann solver. This is accomplished by introducing a constant resolved state between the states being considered, which introduces sufficient dissipation for systems of conservation laws. Closed form expressions for the resolved fluxes are also provided to facilitate numerical implementation. The Riemann solver is proved to be positively conservative for the density variable; the positivity of the pressure variable has been demonstrated for Euler flows when the divergence in the fluid velocities is suitably restricted so as to prevent the formation of cavitation in the flow. We also focus on the construction of multidimensionally upwinded electric fields for divergence-free magnetohydrodynamical flows. A robust and efficient second order accurate numerical scheme for two and three dimensional Euler and magnetohydrodynamic flows is presented. The scheme is built on the current multidimensional Riemann solver. The number of zones updated per second by this scheme on a modern processor is shown to be cost competitive with schemes that are based on a one-dimensional Riemann solver. However, the present scheme permits larger timesteps.
2) Multidimensional HLLE Riemann Solver
The multidimensional HLLE solver models four quadrant states with a single constant resolved state and extremal wave speeds, producing multidimensionally upwinded fluxes. It preserves key one-dimensional and supersonic limits, supports positivity under stated conditions, and contributes stabilizing transverse terms to face fluxes.
- Derivation: The solver formulates a four-state edge Riemann problem on Cartesian or logically rectangular meshes using extremal right-, left-, upward-, and downward-going wave speeds.The four quadrant states meet at an edge, and the speeds bound waves propagating from that edge.
- Derivation: A single constant resolved state between the initial states supplies multidimensional dissipation while retaining closed-form resolved fluxes.The multidimensional wave model is chosen to circumscribe the actual waves and provide adequate dissipation.
- Resolved state: When variations are one-dimensional, the resolved state reduces to the familiar one-dimensional HLLE state; otherwise it incorporates transverse multidimensional variations.The same reduction occurs for variations confined to the y-direction.
- Resolved fluxes: Transverse flux contributions are included in the resolved flux, but strong oblique shocks require a shock detector to suppress them because the flux lacks corresponding transverse dissipation.A flow-dependent parameter β is set to zero near shocks and to one elsewhere.
- Limits and comparisons: The solver retrieves correct upwinded limits in supersonic cases, unlike certain prior multidimensional formulas whose upwind terms differ from careful multidimensional analysis.The paper also notes that one alternative performs strongly on strong supersonic shocks but lacks enhanced dissipation in some subsonic and transonic problems.
- Positivity: The Appendix establishes positive density for Euler and MHD flows and positive pressure for Euler flows when velocity divergence is restricted to prevent cavitation.The pressure result is conditional on the stated restriction on divergent velocities.
- Flux assembly: Multidimensional corner Riemann problems add cross-term contributions to face fluxes, with larger effects as the timestep increases, especially for subsonic flows.These multidimensional contributions have a beneficial and stabilizing effect on the one-dimensional face flux.
4) Brief Description of the One-Step, Second Order Accurate, Predictor-Corrector Scheme for Euler and MHD Flow
The scheme combines an inherently two-dimensional Riemann solver with predictor-corrector timestepping, divergence-free magnetic reconstruction, and limiter-based accuracy control for Euler and MHD flows.
- Multidimensional Riemann solver: The solver operates at edges, producing two transverse upwinded fluxes that are assembled into face-centered fluxes.In MHD, edge solves directly provide the electric field; face fluxes are obtained from the stated flux relations.
- Predictor-corrector scheme: Each timestep calls the multidimensional Riemann solver twice per edge, resulting in six solver calls per zone update.The predictor-corrector scheme is organized into six functional substeps.
- MHD reconstruction: Magnetic fields are reconstructed from face-centered components using transverse limiting to obtain a second-order accurate divergence-free reconstruction.This reconstruction is part of the timestep procedure for MHD calculations.
- Implementation cost: 136,500 and 80,000 three-dimensional zones per second were updated for Euler and MHD, versus 138,600 and 81,250 with one-dimensional HLLE solvers.The multidimensional implementation has only a slight speed difference from the comparable one-dimensional implementation.
- Accuracy and reconstruction: TVD-based minmod and MC schemes fail to meet design accuracy in the L∞ norm, while WENO succeeds for the unmagnetized and magnetized vortex analyses.The reported accuracy analyses measure density for the unmagnetized vortex and the x-component of magnetic field for the magnetized vortex.
6) Multidimensional Test Problems for Euler Flow
The Euler tests cover multidimensional Riemann problems and double Mach reflection, while the broader test suite also includes demanding MHD configurations. The reported solutions capture expected structures and remain positive in the rotor test.
- Euler test setup: The Euler test suite uses CFL number 0.65 and ratio of specific heats 1.4 for multidimensional Riemann and double Mach reflection problems.The tests include multidimensional Riemann problems and the double Mach reflection problem.
- Multidimensional Riemann problems: The first multidimensional Riemann problem produces two Mach reflections and two contact surfaces, with a pronounced density valley moving to the four-shock intersection.The density behavior is reported as consistent with expectations.
- Multidimensional Riemann problems: The double Mach reflection problem uses a 400×400 mesh and yields a mushroom cap captured crisply by the multidimensional Riemann solver.The shock propagates at 45° to the mesh.
- MHD test problems: The field-loop advection test shows virtually no diffusion after one circuit and remains stable with CFL number as large as 0.95.The reported simulation used WENO and CFL number 0.65.
8) Conclusions
The paper presents a multidimensional HLLE solver with a single constant resolved state, proves key positivity properties under stated conditions, and applies it to upwinded MHD electric fields. Its second-order Euler and MHD implementation achieves design accuracy, robust test performance, cost competitiveness, and larger timesteps.
- A single constant resolved state supplies dissipation while closed-form resolved fluxes facilitate implementation.
- The solver preserves positive density for Euler and MHD flows and positive pressure for Euler when velocity divergence excludes cavitation.
- The method produces multidimensionally upwinded electric fields for divergence-free MHD formulations.
- The second-order Euler and MHD scheme meets design accuracy and performs robustly on stringent multidimensional tests.
- The implementation is cost-competitive with one-dimensional-solver schemes while permitting larger timesteps.
Appendix
The Appendix analyzes positivity of the resolved state. Density positivity follows from positive quadrant states, while pressure positivity is supported computationally only under restrictions on multidimensional divergence and cavitation.
- The positivity argument uses extremal wave speeds and focuses on subsonic flow conditions.
- The resolved density is positive when the four quadrant densities are positive definite.
- Pressure positivity is harder to establish because multidimensional cavitation-preventing velocity ranges lack closed-form conditions, leaving a non-ironclad computational demonstration.
- The pressure analysis restricts positive velocity divergence, which measures multidimensional rarefaction rather than collisions.
- With φ ≤ 0.43, pressures remained positive while six density and pressure ratios independently spanned [0.1, 102.4].
- Negative pressures occurred in 0.52% and 1.15% of tested cases for φ = 0.75 and 0.9, respectively.
SRT SLT
The two-dimensional Riemann problem places four quadrant states around a shared edge and tracks waves propagating outward at extremal speeds. Their influence is bounded by a rectangle in space-time.
- Four neighboring zones occupy the four quadrants around the shared edge at the origin of the xy-plane.
- The quadrant states use RU, LU, LD, and RD labels corresponding to right-up, left-up, left-down, and right-down.
- At t=T, waves reach x = S_R T and x = S_L T along x, and y = S_U T and y = S_D T along y.
- The rectangle QMNR bounds the space-time domain affected by the waves.
D F HLL
The multidimensional solver models waves in space-time, combines corner and face Riemann problems into averaged fluxes, and applies these ideas to Euler and MHD calculations. Corner contributions grow with timestep, especially for subsonic cases, while the reported tests remain robust.
- The wave model encloses the actual multidimensional waves and supplies adequate dissipation, including supersonic cases.
- Fluxes through a zone face are assembled from distinct resolved and directional contributions over different portions of the face.
- As Δt increases, corner Riemann problems contribute increasingly to temporally and spatially averaged face fluxes, especially in subsonic cases.
- The reported Euler and MHD calculations include multidimensional density, pressure, velocity, and magnetic-field diagnostics across stringent test problems.