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Provably Positivity-Preserving Constrained Transport (PPCT) Second-Order Scheme for Ideal Magnetohydrodynamics

Dongwen Pang, Kailiang Wu

Abstract

This paper proposes and analyzes a robust and efficient second-order positivity-preserving constrained transport (PPCT) scheme for ideal magnetohydrodynamics (MHD) on non-staggered Cartesian meshes. The PPCT scheme ensures two critical physical constraints: a globally discrete divergence-free (DDF) condition on the magnetic field and the positivity of density and pressure. The method is inspired by a novel splitting technique from [T.A. Dao, M. Nazarov and I. Tomas, J. Comput. Phys., 508:113009, 2024], which divides the MHD system into an Euler subsystem with steady magnetic fields and a magnetic subsystem with steady density and internal energy. To achieve these structure-preserving properties, the PPCT scheme combines a positivity-preserving (PP) finite volume method for the Euler subsystem with a finite difference constrained transport (CT) method for the magnetic subsystem via Strang splitting. The finite volume method employs a new PP limiter that retains second-order accuracy and enforces the positivity of density and pressure, with rigorous proof provided using the geometric quasilinearization (GQL) approach [K. Wu and C.-W. Shu, SIAM Review, 65:1031-1073, 2023]. For the magnetic subsystem, we develop an implicit finite difference CT method that conserves energy and maintains a globally DDF constraint. This nonlinear system is efficiently solved to machine precision using an iterative algorithm. Since the CT method is unconditionally energy-stable and conserves steady density and internal energy, the PPCT scheme requires only a mild CFL condition for the finite volume method to ensure stability and the PP property. While the focus is on 2D cases for clarity, the extension to 3D is discussed. Several challenging numerical experiments, including highly magnetized MHD jets with high Mach numbers, validate the PPCT scheme's accuracy, robustness, and high resolution.

Provably Positivity-Preserving Constrained Transport (PPCT) Second-Order Scheme for Ideal Magnetohydrodynamics

Abstract

This paper proposes and analyzes a robust and efficient second-order positivity-preserving constrained transport (PPCT) scheme for ideal magnetohydrodynamics (MHD) on non-staggered Cartesian meshes. The PPCT scheme ensures two critical physical constraints: a globally discrete divergence-free (DDF) condition on the magnetic field and the positivity of density and pressure. The method is inspired by a novel splitting technique from [T.A. Dao, M. Nazarov and I. Tomas, J. Comput. Phys., 508:113009, 2024], which divides the MHD system into an Euler subsystem with steady magnetic fields and a magnetic subsystem with steady density and internal energy. To achieve these structure-preserving properties, the PPCT scheme combines a positivity-preserving (PP) finite volume method for the Euler subsystem with a finite difference constrained transport (CT) method for the magnetic subsystem via Strang splitting. The finite volume method employs a new PP limiter that retains second-order accuracy and enforces the positivity of density and pressure, with rigorous proof provided using the geometric quasilinearization (GQL) approach [K. Wu and C.-W. Shu, SIAM Review, 65:1031-1073, 2023]. For the magnetic subsystem, we develop an implicit finite difference CT method that conserves energy and maintains a globally DDF constraint. This nonlinear system is efficiently solved to machine precision using an iterative algorithm. Since the CT method is unconditionally energy-stable and conserves steady density and internal energy, the PPCT scheme requires only a mild CFL condition for the finite volume method to ensure stability and the PP property. While the focus is on 2D cases for clarity, the extension to 3D is discussed. Several challenging numerical experiments, including highly magnetized MHD jets with high Mach numbers, validate the PPCT scheme's accuracy, robustness, and high resolution.
Paper Structure (18 sections, 14 theorems, 135 equations, 16 figures, 3 tables)

This paper contains 18 sections, 14 theorems, 135 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Compressible Ideal MHD Equations and Operator Splitting
  3. Two-Dimensional PPCT Scheme
  4. Provably PP Finite Volume Method $S_{A}^{{\Delta t}}$ for System-A
  5. Outline of the Finite Volume Method
  6. Second-Order Reconstruction
  7. Positivity-Preserving Goals
  8. Positivity-Preserving Limiter for Point Values
  9. Proof of Positivity-Preserving Property for Updated Cell Averages
  10. Finite Difference CT Method $S_{B}^{\Delta t}$ for System-B
  11. Finite Difference CT Method and Its Properties
  12. Unique Solvability of System \ref{['eq:nonlinears']} and Convergence Analysis of Algorithm \ref{['iter']}
  13. Algorithmic Flow of PPCT Scheme
  14. Three-Dimensional PPCT Scheme
  15. 3D Provably PP Finite Volume Method $S_{A}^{{\Delta t}}$ for System-A
  16. ...and 3 more sections

Key Result

Lemma 3.4

\newlabelLemma:GQLrepresentation0 The admissible state set $\mathcal{G}$ is exactly equivalent to where ${\bm n}_1=(1,0,0,0,0)^\top$ and $\boldsymbol{n}_* = \left(\frac{ \left| {\bm v}_* \right|^2}{2}, -{\bm v}_*, 1\right)^\top$, with $\boldsymbol{v}_*$ as free auxiliary variables independent of $\bm{Q}$.

Figures (16)

  • Figure 1: Illustration of the provably PP finite volume method with forward Euler time discretization.
  • Figure 1: \ref{['Ex:vortex']} : Iteration error varies with the number of iterations in the case of $C_{\tt CFL} = \frac{2}{3}$ on the $1024 \times 1024$ mesh.
  • Figure 2: \ref{['Ex:OT']}: Density of Orszag-Tang turbulence problem at $t = 2$ (left) and $t = 4$(right). Twenty-four contour lines are displayed.
  • Figure 3: \ref{['Ex:OT']}:Profiles of density (left) and thermal pressure (right) along line $y=0.625 \pi$ at $t = 3$.
  • Figure 4: \ref{['Ex:Rotor']}: The density (top-left), thermal pressure (top-right), magnetic pressure (bottom-left), and Mach number (bottom-right) for the rotor problem at $t = 0.295$.
  • ...and 11 more figures

Theorems & Definitions (36)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4: GQL Representation WuShu2021GQL
  • Theorem 3.5
  • Proof 1
  • Remark 3.6
  • Remark 3.7
  • Theorem 3.8
  • Proof 2
  • ...and 26 more