In this paper we present three different numerical approaches to account for curl-type involution constraints in hyperbolic partial differential equations for continuum physics. All approaches have a direct analogy to existing and well-known divergence-preserving schemes for the Maxwell and MHD equations. The first method consists in a generalization of the Godunov-Powell terms, which means adding suitable multiples of the involution constraints to the PDE system in order to achieve the symmetric Godunov form. The second method is an extension of the generalized Lagrangian multiplier (GLM) approach of Munz et al., where the numerical errors in the involution constraint are propagated away via an augmented PDE system. The last method is an exactly involution preserving discretization, similar to the exactly divergence-free schemes for the Maxwell and MHD equations, making use of appropriately staggered meshes. We present some numerical results that allow to compare all three approaches with each other.
In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of Munz et al. (JCP 2000, JCP 2002), which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type involutions, to the case of hyperbolic PDE systems with curl-type involutions. The key idea here is to solve an augmented PDE system, in which curl errors propagate away via a Maxwell-type evolution system. The new approach is first presented on a simple model problem, in order to explain the basic ideas. Subsequently, we apply it to a strongly hyperbolic f…
Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first order reductions of the Einstein field equations, or a novel first order hyperbolic reformulation of Schrödinger's equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector fie…
On the involution preserving properties of semi-discrete Active Flux methods on simplex
Institute of Mathematics, University of Zürich（苏黎世大学数学研究所）；University of Trento（特伦托大学）；University of Science and Technology, Shenzhen（深圳理工大学）；INRIA, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（法国国家信息与自动化研究所、波尔多大学、法国国家科学研究中心、波尔多理工学院、IMB）
This paper examines a class of PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving respectively. In this paper we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, have to be collocated at the edges of the mesh. They are updated by using potentials that are collocated at the vertices of the mesh. The resulting schemes : …
A simple and general framework for the construction of exactly div-curl-grad compatible discontinuous Galerkin finite element schemes on unstructured simplex meshes
In this paper, we present a semi-implicit numerical solver for a first order hyperbolic formulation of two-phase flow with surface tension and viscosity. The numerical method addresses several complexities presented by the PDE system in consideration: (i) The presence of involution constraints of curl type in the governing equations requires explicit enforcement of the zero-curl property of certain vector fields (an interface field and a distortion field); the problem is eliminated entirely by adopting a set of compatible curl and gradient discrete differential operators on a staggered grid, a…
In this work, we introduce two novel reformulations of a recent weakly hyperbolic model for two-phase flow with surface tension. In the model, the tracking of phase boundaries is achieved by using a vector interface field, rather than a scalar tracer, so that the surface-force stress tensor can be expressed as an algebraic function of the state variables, without requiring the computation of gradients of the tracer. An interesting and important feature of the model is that this interface field obeys a curl involution constraint, that is, the vector field is required to be curl-free at all time…
The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a …
This paper is devoted to the theoretical and numerical study of an optimal design problem in high-temperature superconductivity (HTS). The shape optimization problem is to find an optimal superconductor shape which minimizes a certain cost functional under a given target on the electric field over a specific domain of interest. For the governing PDE-model, we consider an elliptic curl-curl variational inequality (VI) of the second kind with an L1-type nonlinearity. In particular, the non-smooth VI character and the involved H(curl)-structure make the corresponding shape sensitivity analysis ch…
We propose a new pressure-based structure-preserving (SP) and quasi asymptotic preserving (AP) staggered semi-implicit finite volume scheme for the unified first order hyperbolic formulation of continuum mechanics. The unified model is based on the theory of symmetric-hyperbolic and thermodynamically compatible (SHTC) systems and includes the description of elastic and elasto-plastic solids in the nonlinear large-strain regime as well as viscous and inviscid heat-conducting fluids, which correspond to the stiff relaxation limit of the model. In the absence of relaxation source terms, the homog…
Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions（索邦大学，巴黎第七大学SPC，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Université Côte d’Azur, CNRS, Laboratoire J-A Dieudonné（蔚蓝海岸大学，法国国家科学研究中心，J-A 迪厄多内实验室）；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）
A structure-preserving staggered semi-implicit four-split finite volume scheme for continuum mechanics on unstructured meshes
University of Trento（特伦托大学）；Southern University of Science and Technology（南方科技大学）；Université Savoie Mont Blanc（萨瓦-蒙布朗大学）；University of Ferrara（费拉拉大学）；University of Verona（维罗纳大学）；Inria（法国国家数字研究院）；Université de Strasbourg（斯特拉斯堡大学）
