Semi-implicit Finite-Difference Methods for Compressible Gas Dynamics with Curved Boundaries: A Ghost-Point Approach

We consider weakly coupled systems of nonlinear hyperbolic conservation laws on moving surfaces. As in the Euclidean space, see, for example, (Levy, Commun Partial Differ Equ 17(3–4):657–698, 1992, [9], Rohde, Weakly coupled systems of hyperbolic conservation laws. PhD thesis, Mathematische Fakultat der Albert-Ludwigs-Universitat Freiburg 1996, [16]), the coupling is realized by a source term, which only depends on (x, t) and the unknown function u(x, t) but not on its derivatives. Scalar conservation laws on moving surfaces were considered in Dziuk et al., Interfaces Free Boundaries 15:202–236, 2013, [4]), Lengeler and Muller (J Differ Equ 254(4):1705–1727, 2013, [10]). The velocity of the surface is given by a smooth function, and we assume the surface to be compact. We prove the existence for an entropy solution. First, we consider the regularized parabolic problem with viscosity parameter \(\varepsilon \) and show that there exists a weak solution by decoupling and linearizing the problem. Then, we prove the boundedness of this solution in \(L^\infty (G_T)\), use standard regularity results to prove that this solution is a solution in the classical sense, and show uniform boundedness in \(L^\infty (G_T)\) and \(W^{1,1}(G_T)\) with respect to \(\varepsilon \).

This paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions. The main assumption is that the system under study admits a convex entropy extension. It is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either Godunov schemes or by suitable viscous approximations. Furthermore, in the case that the weak interior solutions are generated by Godunov schemes, any Lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values. This in particular gives a trace formula for the flux functions in terms of the numerical boundary data. We also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws. It is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary Riemann problems, and the converse is not true in general. The general theory is then applied to some specific examples. First, several new facts are obtained for convex scalar conservation laws. For example, we give example which show that Godunov schemes produce numerical boundary layers. It is shown that any continuous functions of density have traces on the boundary (instead of only entropy fluxes). We also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data. A generalized Oleinik entropy condition is also obtained. Next, we prove the existence of a weak solution to the initial-boundary value problem for a family of × quadratic system with a uniformly characteristic boundary condition.

We present a class of high‐order weighted essentially nonoscillatory (WENO) reconstructions based on relaxation approximation of hyperbolic systems of conservation laws. The main advantage of combining the WENO schemes with relaxation approximation is the fact that the presented schemes avoid solution of the Riemann problems due to the relaxation approach and high‐resolution is obtained by applying the WENO approach. The emphasis is on a fifth‐order scheme and its performance for solving a wide class of systems of conservation laws. To show the effectiveness of these methods, we present numerical results for different test problems on multidimensional hyperbolic systems of conservation laws. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

We present an overview on some recent works in collaboration with S. Bianchini (see Bianchini and Modena in Lagrangian representation for solution to general systems of conservation laws [9] and the Ph.D. thesis Modena in Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for hyperbolic systems of conservation laws [15]), in which we propose a way to describe BV solutions to hyperbolic systems of conservation laws in one space dimension from a Lagrangian point of view.

This note analyzes a simple discontinuous solution to hyperbolic 2 × 2 systems of conservation laws having quadratic flux functions with an isolated umbilic point where the characteristic speeds are equal. We study the Hugoniot curves in Schaeffer & Shearer's case I and II which are relevant to the three-phase Buckley–Leverett model for oil reservoir flow. The compressive and overcompressive parts are determined. The wave curves through the umbilic point are discussed and their compressive and overcompressive parts are also determined.

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

Chapter 2 - Hyperbolic systems of conservation laws

This paper contains several recent results about nonlinear systems of hyperbolic conservation laws obtained through the technique of Wave Front Tracking.

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles. From the reviews of the 3rd edition: This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject. Evgeniy Panov, Zentralblatt MATH A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the Bible on the subject. Philippe G. LeFloch, Math. Reviews

This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.

Communications on Pure and Applied MathematicsVolume 30, Issue 6 p. 767-796 Article Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws Tai-Ping Liu, Tai-Ping Liu University of MarylandSearch for more papers by this author Tai-Ping Liu, Tai-Ping Liu University of MarylandSearch for more papers by this author First published: November 1977 https://doi.org/10.1002/cpa.3160300605Citations: 90AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume30, Issue6November 1977Pages 767-796 RelatedInformation

In this paper, we establish the existence theory for general system of hyperbolic conservation laws and obtain the uniform L1 boundness for the solutions. The existence theory generalizes the classical Glimm theory for systems, for which each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax. We construct the solutions by the Glimm scheme through the wave tracing method. One of the key elements is a new way of measuring the potential interaction of the waves of the same characteristic family involving the angle between waves. A new analysis is introduced to verify the consistency of the wave tracing procedure. The entropy functional is used to study the L1 boundedness.

We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem. We discuss both approaches in the case of the coupling of two fluid models at a material contact discontinuity, the models being the usual gas dynamics equations with different equations of state. We also study the coupling of two-temperature plasma fluid models and illustrate the approach by numerical simulations.

A scheme for the numerical solution of hyperbolic systems of conservation laws