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An active-flux-type scheme for ideal MHD with provable positivity and discrete divergence-free property

Mengqing Liu, Dongwen Pang, Remi Abgrall, Kailiang Wu

Abstract

We develop a positivity-preserving (PP) PAMPA (Point-Average-Moment PolynomiAl-interpreted) scheme that enforces a discrete divergence-free (DDF) magnetic field for ideal MHD on Cartesian grids. Extending our 1D invariant-domain-preserving (IDP) PAMPA framework (Abgrall, Jiao, Liu, Wu, SIAM J. Sci. Comput., to appear) to multidimensional, multiwave MHD, the method combines a limiter-free PP update of interface point values via a new nonconservative reformulation with a local DDF projection. Cell averages are provably PP under a mild a~priori positivity condition on one cell-centered state, using: (i) DDF-constrained interface values, (ii) a PP limiter only at the cell center, (iii) a PP flux with appropriate wave-speed bounds, and (iv) a suitable discretization of the Godunov--Powell source term. The PP proof employs geometric quasi-linearization (GQL; Wu & Shu, SIAM Review, 2023), which linearizes the pressure constraint. The scheme avoids explicit polynomial reconstructions, is compatible with arbitrarily high-order strong-stability-preserving (SSP) time integration, and is simple to implement. Robustness and resolution are enhanced by a problem-independent Lax-type entropy troubled-cell indicator using only two characteristic speeds and a convex oscillation elimination (COE) mechanism with a new intercell-difference norm. Tests -- including a blast wave with plasma $β\approx 2.51\times 10^{-6}$ and jets up to Mach $10^{4}$ -- show high-order accuracy, sharp MHD-structure resolution, and strong-shock robustness. To our knowledge, this is the first active-flux-type ideal-MHD method rigorously PP for both cell averages and interface point values while maintaining DDF throughout.

An active-flux-type scheme for ideal MHD with provable positivity and discrete divergence-free property

Abstract

We develop a positivity-preserving (PP) PAMPA (Point-Average-Moment PolynomiAl-interpreted) scheme that enforces a discrete divergence-free (DDF) magnetic field for ideal MHD on Cartesian grids. Extending our 1D invariant-domain-preserving (IDP) PAMPA framework (Abgrall, Jiao, Liu, Wu, SIAM J. Sci. Comput., to appear) to multidimensional, multiwave MHD, the method combines a limiter-free PP update of interface point values via a new nonconservative reformulation with a local DDF projection. Cell averages are provably PP under a mild a~priori positivity condition on one cell-centered state, using: (i) DDF-constrained interface values, (ii) a PP limiter only at the cell center, (iii) a PP flux with appropriate wave-speed bounds, and (iv) a suitable discretization of the Godunov--Powell source term. The PP proof employs geometric quasi-linearization (GQL; Wu & Shu, SIAM Review, 2023), which linearizes the pressure constraint. The scheme avoids explicit polynomial reconstructions, is compatible with arbitrarily high-order strong-stability-preserving (SSP) time integration, and is simple to implement. Robustness and resolution are enhanced by a problem-independent Lax-type entropy troubled-cell indicator using only two characteristic speeds and a convex oscillation elimination (COE) mechanism with a new intercell-difference norm. Tests -- including a blast wave with plasma and jets up to Mach -- show high-order accuracy, sharp MHD-structure resolution, and strong-shock robustness. To our knowledge, this is the first active-flux-type ideal-MHD method rigorously PP for both cell averages and interface point values while maintaining DDF throughout.
Paper Structure (23 sections, 3 theorems, 107 equations, 17 figures, 2 tables)

This paper contains 23 sections, 3 theorems, 107 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Structure-Preserving PAMPA Method for Ideal MHD
  4. Framework and key components of structure-preserving PAMPA method
  5. Automatically PP approach for point-value evolution
  6. Provably PP approach for cell-average evolution
  7. Step1: DDF projection
  8. Step 2: PP limiter for cell-center point value
  9. Step 3: Convex decomposition of cell average
  10. Step 4: COE procedure
  11. PP numerical flux
  12. Discretization of the Godunov--Powell source term
  13. Proof of PP property for updated cell averages
  14. A Lax-type entropy criterion as shock indicator
  15. COE: Convex Oscillation Elimination
  16. ...and 8 more sections

Key Result

Theorem 3.1

Assume that $\overline{\bm U}^{\,n}_{i,j} \in \mathcal{G}$ and conditions eq:PPcon--eq:CAD3 hold. Then the updated cell averages produced by EulerForwardCellAvg are PP; i.e., under the CFL condition

Figures (17)

  • Figure 1: The nine degrees of freedom (DoFs) in the third-order PAMPA method include the cell average, point values at edge centers, and point values at cell vertices.
  • Figure 2: The flowchart of our structure-preserving PAMPA scheme for ideal MHD.
  • Figure 3: Illustration of the limiting values produced after the DDF projection ( Step 1), the PP limiter ( Steps 2 and 3), and the COE procedure ( Step 4).
  • Figure 4: \ref{['Ex:OT']}: Density contours for the Orszag--Tang problem at $t = 3$ (left) and $t = 4$ (right). Twenty-four contour levels are shown. Top: without COE; bottom: with COE.
  • Figure 5: \ref{['Ex:Rotor']}: The density (top-left), thermal pressure (top-right), magnetic pressure (bottom-left), and velocity magnitude (bottom-right) for the rotor problem at $t = 0.295$.
  • ...and 12 more figures

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Theorem 3.1: PP Cell Average Update
  • Remark 3
  • Remark 4
  • Theorem 3.2: Scale Invariance
  • proof
  • Remark 5
  • Theorem 3.3: Evolution Invariance
  • proof
  • ...and 8 more