Decoupled iterative schemes for solving stationary MHD problems

Technical Summary: Decoupled Iterative Schemes for Stationary MHD Problems

Problem Statement
The paper addresses the numerical solution of the stationary, incompressible magnetohydrodynamics (MHD) system. This system models the interaction between electrically conducting fluids and electromagnetic fields, governed by the coupling of the Navier-Stokes equations and Maxwell's equations. The mathematical formulation involves finding the velocity field ($u$), magnetic field ($B$), modified pressure ($p$), and a Lagrange multiplier ($\lambda$) satisfying a set of coupled partial differential equations. The primary challenge lies in the strong nonlinearity and the tight coupling between the fluid and magnetic variables, which complicates the construction of efficient and stable numerical solvers.

Methodology
The authors propose and analyze two novel decoupled iterative schemes inspired by the Incremental Viscosity Splitting (IVS) strategy, originally developed for unsteady Navier-Stokes flows.

1. Standard IVS Scheme (Algorithm 5.1):

   • Splitting Strategy: The method splits the momentum and induction equations with respect to their diffusive terms.
   • Step 1: Solves for intermediate variables ($\tilde{u}, \tilde{B}$) by treating the convective and coupling terms explicitly using values from the previous iteration, while retaining the pressure and Lagrange multiplier from the previous step. This results in a coupled linear system for the intermediate fields.
   • Steps 2 & 3: Corrects the intermediate fields to obtain the final velocity ($u$) and magnetic field ($B$) by solving two separate, decoupled Stokes-type systems. These systems enforce the divergence-free constraints ($\nabla \cdot u = 0, \nabla \cdot B = 0$) and update the pressure and Lagrange multiplier.
   • Linear Algebra Advantage: A key feature is that the Schur complement matrices for the Stokes corrections are independent of the iteration level ($k$). Consequently, they need to be assembled and preconditioned only once, reducing computational overhead compared to methods where the system matrix changes at every step.

2. Elsässer IVS Scheme (Algorithm 6.1):

   • Reformulation: To further decouple the system, the authors reformulate the MHD equations using Elsässer variables ($w = u + \sqrt{\kappa}B$ and $z = u - \sqrt{\kappa}B$).
   • Decoupling: In this formulation, the coupling between the new variables occurs through one nonlinear and one linear term. The proposed algorithm solves for $w$ and $z$ in separate steps, replacing the large coupled system of the standard IVS with two smaller, fully decoupled linear systems.
   • Applicability: This approach is theoretically advantageous under homogeneous Dirichlet boundary conditions, though the authors note challenges arise with mixed boundary conditions.

Theoretical Contributions
The paper provides rigorous mathematical analysis for both schemes:

• Well-posedness: The authors establish existence and uniqueness results for the continuous MHD problem under a "small data" assumption (specifically, a bound on the Reynolds and magnetic Reynolds numbers relative to viscosity).
• Convergence Proofs:
  • The classical Picard iterative scheme is analyzed and proven to converge under the same small data assumption.
  • Both the Standard IVS and Elsässer IVS schemes are proven to be uniformly bounded and geometrically convergent to the unique solution.
  • The proofs utilize energy estimates, the inf-sup (LBB) condition, and specific lemmas regarding the convergence of non-negative sequences.
• Error Analysis: Theoretical bounds are derived showing that the errors in velocity, magnetic field, pressure, and Lagrange multipliers decay to zero as the iteration count increases.

Numerical Results
The authors validate their theoretical findings using the FreeFem++ software with $P_2-P_1$ finite element pairs.

• Convergence Tests: Manufactured solutions in 2D demonstrate optimal convergence rates ($O(h^2)$ in $L^2$ and $O(h)$ in $H^1$) for both velocity and magnetic fields, confirming the accuracy of the schemes.
• Benchmark Problems:
  • Hartmann Flow: Both 2D and 3D Hartmann flow problems were solved. The numerical results matched analytical solutions with high accuracy for various Hartmann numbers ($Ha$).
  • Lid-Driven Cavity: The schemes successfully simulated complex flow patterns for various Reynolds ($Re$), magnetic Reynolds ($Rm$), and coupling ($\kappa$) numbers, including high coupling cases ($\kappa=5000$).
  • Flow Over a Step: The method captured flow separation and pressure drops accurately, demonstrating robustness in handling geometric singularities and varying coupling strengths.

Significance and Claims
The paper claims that the proposed decoupled iterative schemes offer a practical and efficient alternative to classical monolithic or fully coupled iterative methods (such as Newton or standard Picard methods).

• Efficiency: By splitting the diffusive terms, the method reduces the complexity of the linear systems solved at each step. Specifically, the Elsässer formulation allows for the solution of two smaller, decoupled systems rather than one large coupled system.
• Reusability: The IVS approach ensures that the Schur complement matrices for the Stokes corrections are static, allowing for a single assembly and preconditioning step, which is computationally beneficial for 3D problems.
• Robustness: The schemes maintain the divergence-free property of the velocity and magnetic fields at the discrete level and remain stable under the established small data conditions.

The authors conclude that while the current work focuses on stationary problems, the methodology provides a foundation for future extensions to time-dependent (unsteady) MHD problems, with a focus on temporal and spatial stability analysis.