Hyperbolic systems under nonconservative form arise in numerous applications modeling physical processes, for example from the relaxation of more general equations (e.g. with dissipative terms). This paper reviews an existing class of augmented Roe schemes and discusses their application to linear nonconservative hyperbolic systems with source terms. We extend existing augmented methods by redefining them within a common framework which uses a geometric reinterpretation of source terms. This results in intrinsically well-balanced numerical discretizations. We discuss two equivalent formulations: (1) a nonconservative approach and (2) a conservative reformulation of the problem. The equilibrium properties of the schemes are examined and the conditions for the preservation of the well-balanced property are provided. Transient and steady state test cases for linear acoustics and hyperbolic heat equations are presented. A complete set of benchmark problems with analytical solution, including transient and steady situations with discontinuities in the medium properties, are presented and used to assess the equilibrium properties of the schemes. It is shown that the proposed schemes satisfy the expected equilibrium and convergence properties.
Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems
Laboratoire de Mathématiques Jean Leray, Centre National de la Recherche Scientifique, Université de Nantes（让·勒雷数学实验室，法国国家科学研究中心，南特大学）；Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室，法国国家科学研究中心，皮埃尔和玛丽·居里大学（巴黎第六大学））
The discontinuous Galerkin finite element method (DGFEM) developed by Rhebergen et al. (2008) offers a robust method for solving systems of nonconservative hyperbolic partial differential equations but, as we show here, does not satisfactorily deal with topography in shallow water flows at lowest order (so-called DG0, or equivalently finite volume). In particular, numerical solutions of the space-DG0 discretised one-dimensional shallow water equations over varying topography are not truly `well-balanced'. A numerical scheme is well-balanced if trivial steady states are satisfied in the numeric…
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…
Given a first-order nonlinear hyperbolic system of conservation laws endowed with a convex entropy-entropy flux pair, we can consider the class of weak solutions containing shock waves depending upon some small scale parameters. In this Note, we define and derive several classes of entropy-dissipating augmented models, as we call them, which involve (possibly nonlinear) second- and third-order augmentation terms. Such terms typically arise in continuum physics and model the viscosity and capillarity effects in a fluid, for instance. By introducing a new notion of positive entropy production th…
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) app…
Numerical methods with controlled dissipation for small-scale dependent shocks
Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique Université Pierre et Marie Curie（雅克-路易·利翁斯实验室与法国国家科学研究中心皮埃尔和玛丽·居里大学）；Seminar for Applied Mathematics D-Math Eidgenössische Technische Hochschule HG（苏黎世联邦理工学院应用数学研讨会）
Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces
Institut de Recherche Mathématiques de Rennes, Université de Rennes I（雷恩数学研究所，雷恩第一大学）；Centre de Mathématiques Appliquées & Centre National de la Recherche Scientifique, Ecole Polytechnique（应用数学中心与国家科学研究中心，巴黎综合理工学院）；Laboratoire Jacques–Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室与国家科学研究中心，皮埃尔和玛丽·居里大学（巴黎第六大学））
Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces
Institut de Recherche Mathématiques de Rennes (IRMAR), Université de Rennes 1（雷恩数学研究所，雷恩第一大学）；Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室，法国国家科学研究中心，皮埃尔和玛丽·居里大学（巴黎第六大学））
Mathematical models and numerical simulations are widely used in the field of hemodynamics, representing a valuable resource to better understand physiological and pathological processes. The theory behind the phenomenon is closely related to the study of incompressible flow through compliant thin-walled tubes. The mechanical interaction between blood flow and vessel wall must be properly described by the model. Recent works show the benefits of characterizing the rheology of the vessel wall through a viscoelastic law. Considering the viscous contribution of the wall material and not simply th…
Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China（数学与统计学学院，西安交通大学，西安，陕西 710049，中国）；School of Mathematical Sciences & Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China（数学科学学院及自然科学研究所，上海交通大学，上海 200240，中国）；School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China（数学与统计学学院及电力工程多相流国家重点实验室，西安交通大学，西安，陕西 710049，中国）
We study in this paper three variants of the high-order Discontinuous Galerkin (DG) method with Runge-Kutta (RK) time integration for the induction equation, analysing their ability to preserve the divergence free constraint of the magnetic field. To quantify divergence errors, we use a norm based on both a surface term, measuring global divergence errors, and a volume term, measuring local divergence errors. This leads us to design a new, arbitrary high-order numerical scheme for the induction equation in multiple space dimensions, based on a modification of the Spectral Difference (SD) metho…
