High-Order Finite-Volume Schemes for Magnetohydrodynamics

New high-order finite-volume numerical schemes for the magnetohydrodynamics equations are proposed in two and three dimensions. Two different sets of magnetohydrodynamics equations are considered. The first set is the ideal magnetohydrodynamics system, which assumes that the fluid can be treated as a perfect conductor. The second set is resistive MHD, which involves non-zero resistivity. A high-order central essentially nonoscillatory (CENO) approach is employed, which combines unlimited k-exact polynomial reconstruction with a monotonicity preserving scheme. The CENO schemes, which were originally developed for compressible fluid flow, are applied to the MHD equations, along with two possible control mechanisms for divergence error of the magnetic field. The hyperbolic fluxes are calculated by solving a Riemann problem at each cell interface, and elliptic fluxes are computed through k-exact gradient interpolation where point-wise values of the gradients are required. Smooth test problems and test cases with discontinuities (weak or strong) are considered, and convergence studies are presented for both the ideal and resistive MHD systems. Several potential space physics applications are explored. For these simulations, cubed-sphere grids are used to model the interaction of the solar wind with planetary bodies or their satellites. The basic cubed-sphere grid discretizes a simulation domain between two concentric spheres using six root blocks (corresponding to the six faces of a cube). Conditions describing the atmosphere of the inner body can be applied at the boundary of the inner sphere. For some problems we also need to solve equations within the inner sphere, for which we develop a seven-block cubed-sphere grid where the empty space inside the interior sphere is discretized as a seventh root block. We consider lunar flow problems for which we employ the seven-block cubed-sphere mesh. Ideal MHD is solved between the inner and outer spheres of the grid, and the magnetic diffusion equations are solved within the inner sphere, which represents the lunar interior. Two cases are considered: one is without intrinsic magnetic field, where only a wake is expected without any bow shock forming ahead of the Moon, and the second is with a small dipole moment to model a lunar crustal magnetic anomaly, in which case a small-scale magnetosphere is expected ahead of the region with the magnetic anomaly

Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations.

This paper is concerned with the application of k-exact finite volume methods for compressible Reynolds-Averaged Navier-Stokes computations of flows around aeronautical configurations including the NACA0012, RAE2822, MDA30P30N, ONERA-M6, CRM and DLR-F11. High-order spatial discretisation is obtained with the Weighted Essentially Non-Oscillatory and the Monotone-Upstream Central Scheme for Conservation Laws methods on hybrid unstructured grids in two- and three- dimensions. Schemes of fifth, third and second order comprise the foundation of the analysis, with main findings suggesting that enhanced accuracy can be obtained with at least a third-order scheme. Steady state solutions are achieved with the implicit approximately factored Lower-Upper Symmetric Gauss-Seidel time advancing technique, convergence properties of each scheme are discussed. The Spalart-Allmaras turbulence model is employed where its discretisation with respect to the high-order framework is assessed. A low-Mach number treatment technique is studied, where recovery of accuracy in low speed regions is exemplified. Results are compared with referenced data and discussed in terms of accuracy, grid dependence and computational budget

Un schéma CENO 3D centré sommet pour l’advection

We present a finite-volume scheme with a quadratic reconstruction of the solution fora 3D hyperbolic equation on structured unstructured meshes formed of tetrahedra. This schemeis an adaptation of the CENO scheme to the vertex-centered context.This work was supportedby the ANR project NORMA under grant ANR-19-CE40-0020-01.Nous présentons un schéma volumes-finis avec une reconstruction quadratique de la solution pour une équation de type hyperbolique 3D sur des maillages non-structurés formés de tétraèdres. Ce schéma est une adaptation au contexte centré-sommet du schéma CENO. Ce travail a été financé par l’Agence Nationale de la Recherche dans le cadre du projet NORMA, contrat ANR-19-CE40-0020-01

Low-Mach number treatment for Finite-Volume schemes on unstructured meshes

The paper presents a low-Mach number (LM) treatment technique for high-order, Finite-Volume (FV) schemes for the Euler and the compressible Navier–Stokes equations. We concentrate our efforts on the implementation of the LM treatment for the unstructured mesh framework, both in two and three spatial dimensions, and highlight the key differences compared with the method for structured grids. The main scope of the LM technique is to at least maintain the accuracy of low speed regions without introducing artefacts and hampering the global solution and stability of the numerical scheme. Two families of spatial schemes are considered within the k-exact FV framework: the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and the Weighted Essentially Non-Oscillatory (WENO). The simulations are advanced in time with an explicit third-order Strong Stability Preserving (SSP) Runge–Kutta method. Several flow problems are considered for inviscid and turbulent flows where the obtained solutions are compared with referenced data. The associated benefits of the method are analysed in terms of overall accuracy, dissipation characteristics, order of scheme, spatial resolution and grid composition

Adaptation de maillage pour des approximations k-exact en CFD

This paper illustrates the application of error estimates based on k-exactness of approximation schemes for building mesh adaptive approaches able to produce better numerical convergence to continuous solution. The cases of k = 1 and k = 2, i.e. second-order and third-order accurate approximations with steady and unsteady flows are considered.Cet article illustre l’application d’estimations d’erreur, basées sur les schémas d’approximation k-exactitude, pour la construction d’approches adaptatives en maillage permettant de produire une meilleure convergence numérique en solution continue. Les cas de k = 1 et k = 2, c'est-à-dire les approximations précises des deuxième et troisième ordres avec des écoulements stables et instables sont prises en compte