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Divergence-Free Reconstruction of Magnetic Fields and WENO Schemes for Magnetohydrodynamics

Dinshaw S. Balsara

arXiv:0811.2192v1physics.comp-ph

TL;DR

The paper addresses the need for higher-than-second-order accuracy in computational MHD while preserving divergence-free magnetic-field reconstruction. It extends divergence-free reconstruction, designs higher-order WENO schemes with matching Runge–Kutta time integration, and reports better-than-second-order accuracy and successful stringent tests.

  • Problem

    Computational MHD would benefit from schemes with third and higher orders of accuracy while retaining robust divergence-free treatment.

  • Method

    The paper extends divergence-free reconstruction of magnetic fields and combines it with divergence-free WENO schemes and multistage Runge–Kutta time integration.

  • Results

    The resulting schemes achieve better-than-second-order accuracy and perform well on stringent test problems.

  • Takeaways & Limitations

    Higher-order divergence-free schemes provide a route to robust numerical MHD calculations beyond second-order accuracy.

  • Takeaways & Limitations

    Divergence-cleaning schemes show significant inadequacies on sufficiently stringent astrophysical test problems.

Abstract

from arXiv · show

Balsara (2001, J. Comput. Phys., 174, 614) showed the importance of divergence-free reconstruction in adaptive mesh refinement problems for magnetohydrodynamics (MHD) and the importance of the same for designing robust second order schemes for MHD was shown in Balsara (2004, ApJS, 151, 149). Second order accurate divergence-free schemes for MHD have shown themselves to be very useful in several areas of science and engineering. However, certain computational MHD problems would be much benefited if the schemes had third and higher orders of accuracy. In this paper we show that the reconstruction of divergence-free vector fields can be carried out with better than second order accuracy. As a result, we design divergence-free weighted essentially non-oscillatory (WENO) schemes for MHD that have order of accuracy better than second. A multistage Runge-Kutta time integration is used to ensure that the temporal accuracy matches the spatial accuracy. Accuracy analysis is carried out and it is shown that the schemes meet their design accuracy for smooth problems. Stringent tests are also presented showing that the schemes perform well on those tests.

1) Introduction

The paper develops finite-volume, divergence-free WENO schemes for MHD beyond second-order accuracy by extending divergence-free magnetic-field reconstruction and matching spatial and temporal orders. The approach targets accurate, reliable computation for stringent and multiscale MHD problems.

  • Motivation: Violating the divergence-free constraint introduces unphysical plasma transport orthogonal to the magnetic field through extra momentum and energy source terms.The paper motivates divergence-free schemes by linking constraint violations to unphysical transport and added source terms.
  • Motivation: Earlier comparisons found significant inadequacies in divergence-cleaning schemes for sufficiently stringent astrophysical test problems.This motivates robust divergence-free alternatives for numerical MHD.
  • Contribution: The proposed method is a finite-volume divergence-free scheme for MHD that goes beyond second-order accuracy.Finite-volume formulations are emphasized for applications involving non-uniform or adaptive meshes.
  • Method: Divergence-free magnetic-field reconstruction is extended to all orders up to fourth and combined with efficient WENO interpolation for high-order reconstruction.This extends earlier divergence-free reconstruction while retaining a WENO-based spatial reconstruction strategy.
  • Method: An appropriately accurate Runge-Kutta time integrator gives WENO schemes whose spatial and temporal accuracy both exceed second order.The time integration is selected so temporal accuracy matches the higher-order spatial discretization.
  • Evaluation: The paper includes an accuracy analysis and stringent tests to evaluate the resulting higher-order schemes.The introduction identifies smooth-problem accuracy analysis and multiple tests as parts of the evaluation.

2) Higher Order Divergence-Free Reconstruction of Vector Fields

The paper develops divergence-free reconstructions of vector fields beyond second order by adding higher-order moments while enforcing the divergence-free constraint. The construction uses scaled Legendre basis functions and cyclic variable rotations to obtain third- and fourth-order coefficients.

  • The reconstruction framework targets divergence-free vector fields with third- and fourth-order accuracy, extending earlier Cartesian- and logically rectangular-mesh formulations.
  • Scaled Legendre polynomials and tensor-product modal basis functions represent field components within zones and on their faces.
  • Higher-order accuracy is obtained by adding the appropriate facial and interior moments to the second-order representation.
  • Cyclic rotations of variables provide analogous coefficients for the y- and z-components, while the same procedure identifies terms required for second, third, and fourth order.
  • The divergence-free constraint is enforced through coefficient constraints, with higher-order terms determined using matching conditions and SVD minimization of reconstructed magnetic energy.
  • The resulting divergence-free magnetic-field reconstruction helps keep pressure positive in stringent test problems.

3) Step-by-Step Description of the RK-WENO Schemes for Divergence-free MHD

The RK-WENO formulation combines conservative ideal-MHD equations with divergence-free electric-field construction and multistage Runge-Kutta time integration. Its spatial representation uses an efficient WENO scheme for structured meshes.

  • The ideal-MHD equations are written in conservative form with density, momentum, and total-energy variables and ideal fluxes F, G, and H.
  • MHD flux and electric-field symmetries follow from the induction equation and support the construction of edge electric fields from upwinded Godunov fluxes.
  • Avoiding spatial averaging removes the inherent second-order limitation of the earlier Balsara–Spicer electric-field scheme and permits extension to all orders.
  • Once electric fields are available at required zone-edge collocation points, a discrete induction equation can be constructed.
  • The method uses multistage Runge-Kutta time discretization together with an efficient WENO spatial representation for structured meshes.

3.1) Divergence-Free WENO Reconstruction Step

The divergence-free WENO reconstruction step begins by representing field variables within each zone using moments appropriate to the target spatial accuracy at every Runge-Kutta stage.

  • Each finite-volume stage first reconstructs the field variables inside a zone.
  • Including appropriate flow moments yields correspondingly higher spatial accuracy.
  • The reconstruction is repeated as the first task at each stage of the multistage Runge-Kutta scheme.
  • The zone representation is expressed in a modal basis containing constant and higher-order tensor-product terms.

U U U

The reconstruction representation expands the zone-averaged state with modal terms whose included order determines the spatial accuracy.

  • The state representation contains tensor-product basis terms in the local x, y, and z coordinates.
  • Third-order accuracy is indicated by adding the corresponding higher-order modal terms to the representation.

U U U

The reconstruction expression combines directional polynomial contributions to represent the conserved-variable state within the zone.

  • The expression sums polynomial contributions associated with the x, y, and z directions.

U U U

The scheme reconstructs conserved variables and magnetic-field moments using WENO interpolation together with divergence-free magnetic reconstruction.

  • The fourth-order reconstruction includes a larger set of polynomial modes than lower-order schemes.
  • The vector of conserved variables includes zone-averaged mass, momentum, and total-energy densities.
  • The remaining magnetic components are obtained from divergence-free reconstruction using facially averaged magnetic-field components.
  • WENO reconstruction on the faces supplies moments that are transformed into the required zone moments.
  • The method can use several WENO interpolation choices, including strategies developed for structured meshes.

3.2) Flux and Electric Field Evaluation Step

The flux and electric-field evaluation uses reconstructed left and right states, a linearized Riemann solver, face averaging, and edge-based electric-field averaging.

  • Higher-order evaluation requires suitably accurate fluxes and electric fields.
  • The flux is expressed as a linear combination of four vectors, including left and right conserved states and fluxes.
  • A linearized Riemann solver evaluates the flux at general points on a zone boundary.
  • Evaluating the dissipation matrix once at the boundary barycenter freezes the dissipation model across that boundary.
  • Face fluxes are averaged over reference-element coordinates, while edge electric fields combine four face contributions and are then averaged.
  • The resulting fluid fluxes and electric fields are used at each stage of the multistage Runge-Kutta update.

3.3) Multi-Stage Runge Kutta Time Update Step

The method advances the MHD equations with multistage Runge-Kutta schemes, but its fourth-order spatial scheme uses third-order temporal accuracy and requires a reduced Courant number for fourth-order accuracy tests.

  • Strong-stability-preserving Runge-Kutta schemes provide the multistage time integration framework.
  • At each stage, the method computes face fluxes and edge electric-field components before updating the conserved variables.
  • The spatial operator L(U) discretizes the evolution equations used by the Runge-Kutta update.
  • The fourth-order Runge-Kutta scheme was not implemented, so the spatially fourth-order scheme used equation (27) for temporal updates.
  • For fourth-order accuracy demonstrations, the Courant number was reduced by about 0.396 for each doubling of the number of zones.
  • The temporal-order deficiency is identified as being ameliorated by the ADER schemes presented in related work.

4) Accuracy Analysis

The accuracy analysis finds that the divergence-free schemes meet their intended orders, with higher-order schemes achieving comparable accuracy on fewer zones. Fourth-order analysis required smaller meshes because the time integrator used a diminishing Courant number at increasing resolution.

  • The schemes meet their design accuracies in the reported accuracy analyses.
  • 4.1) Magnetized Isodensity Vortex in Two Dimensions: Third-order accuracy at 128x128 zones achieves the same L1 error as second-order accuracy at 256x256 zones.
  • Higher-order schemes provide improved solution quality at the same resolution and require fewer zones to match lower-order accuracy.
  • 4.2) Torsional Alfven Wave Propagation in Three Dimensions: Fourth-order accuracy analysis required a diminishing Courant number as resolution increased, restricting that analysis to smaller meshes.
  • 4.2) Torsional Alfven Wave Propagation in Three Dimensions: On a 16x16x16 mesh, the fourth-order scheme improves over the second-order scheme by more than one order of magnitude.

5) Test Problems

The test problems evaluate the schemes on multidimensional MHD configurations, including wave propagation, rotor dynamics, and a strong low-β shock. Higher-order schemes reduce numerical dissipation and preserve sharp, physically admissible structures in stringent tests.

  • The tests are inherently two-dimensional because divergence-free magnetic-field reconstruction matters in multiple dimensions.
  • Alfven Wave Propagation: Increasing accuracy order substantially reduces numerical dissipation regardless of the Riemann solver used.
  • Alfven Wave Propagation: The linearized Riemann solver improves lower-order schemes more than higher-order schemes, allowing higher-order schemes to use less expensive solvers.
  • Rotor Problem: The fourth-order WENO scheme performs consistently with prior results on the multidimensional rotor problem.
  • Strong Shock Problem: In a low-β plasma with an oblique strong shock, the schemes capture structures crisply and maintain positive pressure.

6) Conclusions

The paper extends divergence-free reconstruction beyond second order and uses it to construct third- and fourth-order divergence-free WENO schemes. Smooth-problem analyses and stringent MHD tests show that the schemes achieve their target accuracy while combining strong-shock capture with low dissipation.

  • Divergence-free vector-field reconstruction can be extended to higher orders.
  • The extension yields third- and fourth-order divergence-free WENO schemes.
  • The schemes meet their design accuracies on smooth test problems.
  • Stringent tests show that the schemes combine strong-shock capture with low dissipation in contact discontinuities and Alfven waves.
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