Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics

Abstract

The magnetohydrodynamics (MHD) equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables, and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems.

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

“…Different control volumes are defined depending on which variable is considered. In Figure 2, we represent in different colors the control volumes corresponding to the primary variables u, v, and p. 1 The pressure unknowns p are defined at the centers of the blocks (marked by ×-points in Figure 2), and the components of the velocity unknowns, u and v, are located at the centers of the block faces (denoted by the •-and •-points in the same figure). For the description of the discrete scheme, we need to fix an adequate indexing for the unknowns, which can be seen in Figure 3, where each unknown is depicted together with the corresponding control volume and the different variables around it.…”

“…The proposed smoother in our work is based on Uzawa relaxation, but different relaxation schemes can be considered. The Braess-Sarazin method [1,5] is an example of another relaxation method. More concretely, the Braess-Sarazin method is based on the matrix system…”

Preconditioning a mass‐conserving discontinuous Galerkin discretization of the Stokes equations

“…An alternative choice that yields less memory use and less time‐per‐iteration is the ‘diagonal’ Vanka submatrix : where F ℓ ℓ , B ℓ ℓ , and are defined as previously. The methods using this submatrix will be called ‘diagonal Vanka’ if we are using the element‐wise blocks and ‘diagonal extended Vanka’ if we are using the extended blocks.…”

“…One strategy is to consider the diagonal of F , C diag =diag( F ) . However, in order to improve robustness and convergence, we also consider a block‐diagonal preconditioner , denoted C blkDiag =blkDiag( F ), in which the blocks correspond to the two velocity degrees of freedom on an edge in the mesh. Note that this choice of blocks is dependent upon the discretization considered.…”

A new multigrid formulation for high order finite difference methods on summation-by-parts form

“…This methodology leads to improved convergence for elliptic problems in a straightforward way [2,3], while partial differential equations with a dominant hyperbolic character are often handled with additional artificial dissipation [4,5]. Multigrid methods are used in various branches of applied mathematics and engineering, such as electromagnetics [6], magnetohydrodynamics [7] and fluid dynamics [8].…”