Second-Order-accurate Schemes for Magnetohydrodynamics with Divergence-free Reconstruction

By Dinshaw S. Balsara. 2003

303 citations recorded by the source corpus.

While working on an adaptive mesh refinement (AMR) scheme for divergence-free magnetohydrodynamics (MHD), Balsara discovered a unique strategy for the reconstruction of divergence-free vector fields. Balsara also showed that for one-dimensional variations in flow and field quantities the reconstruction reduces exactly to the total variation diminishing (TVD) reconstruction. In a previous paper by Balsara the innovations were put to use in studying AMR-MHD. While the other consequences of the invention especially as they pertain to numerical scheme design were mentioned, they were not explored in any detail. In this paper we begin such an exploration. We study the problem of divergence-free numerical MHD and show that the work done so far still has four key unresolved issues. We resolve those issues in this paper. It is shown that the magnetic field can be updated in divergence-free fashion with a formulation that is better than the one in Balsara & Spicer. The problem of reconstructing MHD flow variables with spatially second-order accuracy is also studied. Some ideas from weighted essentially non-oscillatory (WENO) reconstruction, as they apply to numerical MHD, are developed. Genuinely multidimensional reconstruction strategies for numerical MHD are also explored. The other goal of this paper is to show that the same well-designed second-order-accurate schemes can be formulated for more complex geometries such as cylindrical and spherical geometry. Being able to do divergence-free reconstruction in those geometries also resolves the problem of doing AMR in those geometries; the appendices contain detailed formulae for the same. The resulting MHD scheme has been implemented in Balsara's RIEMANN framework for parallel, self-adaptive computational astrophysics. The present work also shows that divergence-free reconstruction and the divergence-free time update can be done for numerical MHD on unstructured meshes. As a result, we establish important analogies between MHD on structured meshes and MHD on unstructured meshes because such analogies can guide the design of MHD schemes and AMR-MHD techniques on unstructured meshes. The present paper also lays out the roadmap for designing MHD schemes for structured and unstructured meshes that have better than second-order accuracy in space and time. All the schemes designed here are shown to be second-order-accurate. We also show that the accuracy does not depend on the quality of the Riemann solver. We have compared the numerical dissipation of the unsplit MHD schemes presented here with the dimensionally split MHD schemes that have been used in the past and found the former to be superior. The dissipation does depend on the Riemann solver, but the dependence becomes weaker as the quality of the interpolation is improved. Several stringent test problems are presented to show that the methods work, including problems involving high-velocity flows in low-plasma-β magnetospheric environments. Similar advances can be made in other fields, such as electromagnetics, radiation MHD, and incompressible flow, that rely on a Stokes-law type of update strategy.

Subfields: Atmospheric Science, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/abcce350c6b78f2ab3747fe412d8e631b93d37a9

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Citations in the corpus

A new class of accurate, mesh-free hydrodynamic simulation methods (2014, background): “As noted by Balsara (2004), the constant β must have a value β > 0.5 in order to maintain the second-order accuracy of the scheme (with lower values being more stable, but also more diffusive).”
Simulating Radiating and Magnetized Flows in Multiple Dimensions with ZEUS-MP (2005, methodology): “Indeed, the past decade has seen the emergence of several newMHDcodes based onGodunov methods (Ryu et al. 1998; Londrillo & Del Zanna 2000, 2004; Balsara 2004; Gardiner & Stone 2005).”
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes (2008, methodology): “We remind the reader that the best error norms obtained in [4] with second order TVD schemes were of the order 10 on 800 × 800 meshes for q = 1 2 , corresponding to an equivalent total number of 512 · 10(6) grid points for a fully 3D computation as presented here.”
An unsplit Godunov method for ideal MHD via constrained transport (2005, methodology): “schemes, such as wave propagation algorithms [19]. The CT 18 algorithms described here can also be applied to integration algorithms based upon spatial reconstruction to the grid cell corners as well [1]. While this might seem surprising following this presentation, note that the factors (δx,δy) can”
An HLLC Riemann solver for relativistic flows – II. Magnetohydrodynamics (2006, unknown): “…δ yq have been computed according to (63) or (65), we apply the correction δx q → τδx q, δyq → τδyq, (A4) where the multidimensional limiter τ is constructed as in Balsara (2004): τ = min [ 1, ψ min ( q̂max − q δmax , q − q̂min δmin )] , (A5) with δmax = max(|δ x q|, |δ yq|), δmin = min(|δ x q|,…”
FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems (2010, unknown): “lowed is analogous to the one described in [23] and [4], where convergence is assessed via the analytical solution of an isentropic vortex.”