The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
Eleuterio F. Toro - One of the best experts on this subject based on the ideXlab platform.
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The Riemann Problem in computational science
2012Co-Authors: Eleuterio F. ToroAbstract:I start by defining the classical Riemann Problem for evolutionary partial differential equations as the simplest Cauchy Problem with piece-wise constant initial conditions, giving examples for hyperbolic systems and for the diffusion equation. A brief historical account of the Riemann Problem for hyperbolic systems is then given, before reviewing the various applications of the Riemann Problem solution, with particular emphasis on the design of a wide variety of numerical methods for solving the relevant general initial-boundary value Problems. Some recent generalizations of the classical Riemann Problem are discussed, in which the initial condition is piecewise smooth and the equations include source terms.
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The Riemann Problem for the Euler Equations
Riemann Solvers and Numerical Methods for Fluid Dynamics, 2009Co-Authors: Eleuterio F. ToroAbstract:In his classical paper of 1959, Godunov [130] presented a conservative extension of the first-order upwind scheme of Courant, Isaacson and Rees [89] to non-linear systems of hyperbolic conservation laws. The key ingredient of the scheme is the solution of the Riemann Problem. The purpose of this chapter is to provide a detailed presentation of the complete, exact solution to the Riemann Problem for the one-dimensional, time-dependent Euler equations for ideal and covolume gases, including vacuum conditions. The methodology can then be applied to other hyperbolic systems.
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The Derivative Riemann Problem: The basis for high order ADER Schemes
2006Co-Authors: Eleuterio F. Toro, Michael Dumbser, Vladimir Titarev, Martin KäserAbstract:The corner stone of arbitrary high order schemes (ADER schemes) is the solution of the derivative Riemann Problem at the element interfaces, a generalization of the classical Riemann Problem first used by Godunov in 1959 to construct a first-order upwind numerical method for hyperbolic systems. The derivative Riemann Problem extends the possible initial conditions to piecewise smooth functions, separated by a discontinuity at the interface. In the finite volume framework, these piecewise smooth functions are obtained from cell averages by a high order non-oscillatory WENO reconstruction, allowing hence the construction of non-oscillatory methods with uniform high order of accuracy in space and time.
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solution of the generalized Riemann Problem for advection reaction equations
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2002Co-Authors: Eleuterio F. Toro, V. A. TitarevAbstract:We present a method for solving the generalized Riemann Problem for partial differential equations of the advection–reaction type. The generalization of the Riemann Problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann Problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann Problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow–water equations with variable bed elevation.
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Solution of the generalized Riemann Problem for advection–reaction equations
Proceedings of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 2002Co-Authors: Eleuterio F. Toro, V. A. TitarevAbstract:We present a method for solving the generalized Riemann Problem for partial differential equations of the advection–reaction type. The generalization of the Riemann Problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann Problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann Problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow–water equations with variable bed elevation.
V. A. Titarev - One of the best experts on this subject based on the ideXlab platform.
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solution of the generalized Riemann Problem for advection reaction equations
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2002Co-Authors: Eleuterio F. Toro, V. A. TitarevAbstract:We present a method for solving the generalized Riemann Problem for partial differential equations of the advection–reaction type. The generalization of the Riemann Problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann Problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann Problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow–water equations with variable bed elevation.
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Solution of the generalized Riemann Problem for advection–reaction equations
Proceedings of the Royal Society of London. Series A: Mathematical Physical and Engineering Sciences, 2002Co-Authors: Eleuterio F. Toro, V. A. TitarevAbstract:We present a method for solving the generalized Riemann Problem for partial differential equations of the advection–reaction type. The generalization of the Riemann Problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann Problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann Problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow–water equations with variable bed elevation.
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The generalised Riemann Problem: the basis of ADER schemes
2002Co-Authors: E.f. Toro, V. A. TitarevAbstract:Advection-reaction type partial differential equations model a wide variety of phenomena in several disciplines in physics, chemistry, environmental sciences, geometry, financial mathematics and many others. Generally, these non-linear inhomogeneous equations must be solved in complicated multi-dimensional domains and thus analytical solutions are only available under very special circumstances. Situations in which exact solutions are available include Riemann Problems. Conventionally, the Riemann Problem for a system of conservation laws in two independent variables and is the initial value Problem for the system with initial conditions consisting of two constant states separated by a discontinuity at the origin ; for background see [3] and [8], for example. Ben-Artzi and Falcovitz [1] and others have generalised the concept of Riemann Problem by admitting initial conditions that are linear functions in , separated by a discontinuity at . We introduce the notation to denote this generalisation of the conventional Riemann Problem, denoted by . Here we extend further the concept of generalised Riemann Problem in two respects, the first concerns initial conditions and the second concerns the type of governing equations. As to initial conditions, we admit -th order polynomial functions of and denote the corresponding generalised Riemann Problem by . The most general case is that in which the initial conditions are two arbitrary but infinitely differentiable functions of , with the corresponding generalised Riemann Problem denoted by . Concerning the governing differential equations, in addition to pure non-linear advection, we include here reaction-like terms; these source terms are assumed to be arbitrary but sufficiently smooth algebraic functions of the unknowns. We propose a semi-analytical method of solution of the generalised Riemann Problem for non-linear advection-reaction partial differential equations. The method gives the solution at at a time , assumed to be sufficiently small, in terms of a time Taylor series expansion at about . The leading term in this expansion is the exact solution of a conventional non-linear Riemann Problem, , with piece-wise constant initial conditions. All remaining terms in the expansion have coefficients that are time derivatives of the solution; these time der ivatives are replaced by spatial derivatives by repeated use of the differential equations, a technique known as the Lax-Wendroff procedure [4]. It is then shown that all spatial derivatives in the expansion obey inhomogeneous advection equations. As derivative values are required at time , all source terms in these evolution equations for all-order spatial derivatives vanish and one only requires the solution of the conventional Riemann Problems with piece-wise constant initial conditions for each of these advection equations. In summary, we reduce the solution of the generalised advection-reaction Riemann Problem to that of solving a sequence of conventional Riemann Problems for homogeneous advection equations, the solutions of which are pieced together to determine the complete solution. The approach can be extended further to solve the general initial value Problem for advection-reaction partial differential equations with piece-wise smooth initial conditions for sufficiently small times. At any position where the initial condition is smooth the solution procedure applies trivially. In the case in which the initial condition includes more than one discontinuity, one places local origins at these positions to define local Problems and applies the solution method as described. In this manner, for a sufficiently short time, one obtains the solution everywhere at any point that is sufficiently far away from points of initial discontinuities. This restriction is necessary to preserve time smoothness in the local Riemann Problems. The solution technique presented here can be used for assessing the performance of numerical methods intended for solving complicated Problems involving both non-linear advection and reaction. Given its local character, our solution procedure has also the potential for providing sub-cell resolution in numerical computation procedures for the general initialboundary value Problem. Finally, our solution procedure can be used to construct numerical methods of very high order of accuracy in space and time to solve the general initial-boundary value Problem; results reported in [9], [11], [6], [7], [10], [5] look very promising. These methods extend existing numerical approaches such as those of Godunov [3] and Glimm [2] in which local solutions of conventional Riemann Problems with piece-wise constant initial conditions are pieced together to advance the solution in time. Finally we make some remarks concerning the generalised Riemann Problem for advectiondiffusion equations.
Wancheng Sheng - One of the best experts on this subject based on the ideXlab platform.
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Global Solutions of the Perturbed Riemann Problem for the Chromatography Equations
Acta Mathematica Scientia, 2019Co-Authors: Ting Zhang, Wancheng ShengAbstract:The Riemann Problem for the chromatography equations in a conservative form is considered. The global solution is obtained under the assumptions that the initial data are taken to be three piecewise constant states. The wave interaction Problems are discussed in detail during the process of constructing global solutions to the perturbed Riemann Problem. In addition, it can be observed that the Riemann solutions are stable under small perturbations of the Riemann initial data.
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The Riemann Problem of conservation laws in magnetogasdynamics
Communications on Pure and Applied Analysis, 2012Co-Authors: Wancheng ShengAbstract:In this paper, we study the Riemann Problem for a simplified model of one dimensional ideal gas in magnetogasdynamics. By using the characteristic analysis method, we prove the global existence of solutions to the Riemann Problem constructively under the Lax entropy condition. The image of contact discontinuity in magnetogasdynamics is a curve with $u=Const.$ in the $(\tau,p,u)$ space. Its projection on the $(p,u)$ plane is a straight line that parallels to the $p$-axis. In contrast with the Problem in gas dynamics, the result causes more complicated and difficult than that in gas dynamics.
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Riemann Problem of a Chapman-Jouguet combustion model for pressure-gradient system
Journal of Shanghai University (English Edition), 2010Co-Authors: Cui Cong, Geng Lai, Wancheng ShengAbstract:In this paper, the Riemann Problem of a Chapman-Jouguet combustion model for the pressure-gradient equations is considered. By analyzing in phase space, existence and uniqueness of the solution to the Riemann Problem are obtained constructively under the global entropy conditions.
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structural stability of solutions to the Riemann Problem for ascalar nonconvex cj combustion model
Discrete and Continuous Dynamical Systems, 2009Co-Authors: Wancheng Sheng, Tong ZhangAbstract:The Riemann Problem for the simplest scalar nonconvex CJ combustion model is considered. A set of entropy conditions are summarized, including pointwise entropy conditions and global entropy conditions. The later is according to the requirement of the structural stability. The unique self-similar entropy solution of the Riemann Problem is constructed case by case. Transition from deflagration to detonation is shown, which dose not occur for the convex model.
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Riemann Problem for a simplest scalar nonconvex ZND combustion model with viscosity
Journal of Shanghai University (English Edition), 2008Co-Authors: Geng Lai, Wancheng ShengAbstract:In this paper, a simplest scalar nonconvex ZND combustion model with viscosity is considered. The existence of the global solution of the Riemann Problem for the combustion model is obtained by using the fixed point theorem.
Lal Singh - One of the best experts on this subject based on the ideXlab platform.
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The generalized Riemann Problem for the Chaplygin gas equation
European Journal of Mechanics - B Fluids, 2020Co-Authors: Pooja Gupta, Rahul Kumar Chaturvedi, Lal SinghAbstract:Abstract The main motive of the present study is to determine the closed form solution of the generalized Riemann Problem (GRP) for the 1-D Euler’s equation for Chaplygin gas by using the Differential Constraint Method (DCM). The consistency conditions and differential constraint for the governing model is determined. Also, the compatibility condition between differential constraints and Chaplygin gas model is computed here. Further, to obtain an exact (closed form) solution of Generalized Riemann Problem (GRP), we characterize the solution for a smooth arbitrary function as initial data. Ultimately, we solve the GRP with generalized discontinuous initial data.
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Solution of generalized Riemann Problem for hyperbolic p−system with damping
International Journal of Non-Linear Mechanics, 2019Co-Authors: Rahul Kumar Chaturvedi, Pooja Gupta, Lal SinghAbstract:Abstract The aim of this paper is to determine the exact solution of the generalized Riemann Problem for the 2 × 2 hyperbolic p − system with linear damping. Differential constraint method is used to find the solution of the system of governing equations. Also, consistency conditions and constraint equations for the hyperbolic system are obtained. Further, we solve the generalized Riemann Problem of non-homogeneous hyperbolic p − system which involves discontinuous initial data.
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Solution of Riemann Problem for ideal polytropic dusty gas
Chaos Solitons & Fractals, 2017Co-Authors: Triloki Nath, R.k. Gupta, Lal SinghAbstract:Abstract The Riemann Problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady flow of an ideal polytropic gas with dust particles is solved analytically without any restriction on magnitude of the initial states. The elementary wave solutions of the Riemann Problem, that is shock waves, rarefaction waves and contact discontinuities are derived explicitly and their properties are discussed, for a dusty gas. The existence and uniqueness of the solution for Riemann Problem in dusty gas is discussed. Also the conditions leading to the existence of shock waves or simple waves for a 1-family and 3-family curves in the solution of the Riemann Problem are discussed. It is observed that the presence of dust particles in an ideal polytropic gas leads to more complex expression as compared to the corresponding ideal case; however all the parallel results remain same. Also, the effect of variation of mass fraction of dust particles with fixed volume fraction (Z) and the ratio of specific heat of the solid particles and the specific heat of the gas at constant pressure on the variation of velocity and density across the shock wave, rarefaction wave and contact discontinuities are discussed.
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Solution of the Riemann Problem in magnetogasdynamics
International Journal of Non-Linear Mechanics, 2014Co-Authors: R. Singh, Lal SinghAbstract:In the present paper, the Riemann Problem for a quasilinear hyperbolic system of equations, governing the one dimensional unsteady flow of an inviscid and perfectly conducting gas, subjected to transverse magnetic field, is solved analytically without any restriction on the initial states. This class of equations includes, as a special case, the Euler equation of gasdynamics. The elementary wave solutions of the Riemann Problem, that is, shock waves, rarefaction waves and contact discontinuities are derived and their properties are discussed. It is noticed that although the magnetogasdynamics system is more complex than the corresponding gasdynamics system, all the parallel results remain identical. It is also assessed as to how the presence of magnetic field influences the variation of velocity and density across the shock wave, rarefaction wave and contact discontinuities.
Hanchun Yang - One of the best experts on this subject based on the ideXlab platform.
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Riemann Problem for the 2d scalar conservation law involving linear fluxes with discontinuous coefficients
Journal of Mathematical Physics, 2020Co-Authors: Hongjun Cheng, Hanchun YangAbstract:This paper is devoted to the four-constant Riemann Problem for the two-dimensional (2D) scalar conservation laws involving linear fluxes with discontinuous coefficients. First, under the assumption that each discontinuity ray of initial data outside of the origin emits exactly one elementary wave, by studying the pointwise interactions occurring at the interaction points of waves, we completely solve this Riemann Problem in the self-similar plane with 21 nontrivial and different geometric structures. Second, when each discontinuity ray of initial data outside of the origin emits two different kinds of contact discontinuities, by studying the pointwise interactions, we construct an interesting kind of spiral structure in the self-similar plane.
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Riemann Problem for the isentropic relativistic Chaplygin Euler equations
Zeitschrift für angewandte Mathematik und Physik, 2012Co-Authors: Hongjun Cheng, Hanchun YangAbstract:This paper studies the Riemann Problem of the isentropic relativistic Euler equations for a Chaplygin gas. The solutions exactly include five kinds. The first four consist of different contact discontinuities while the rest involves delta-shock waves. Under suitable generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.
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Riemann Problem for the relativistic Chaplygin Euler equations
Journal of Mathematical Analysis and Applications, 2011Co-Authors: Hongjun Cheng, Hanchun YangAbstract:Abstract The relativistic Euler equations for a Chaplygin gas are studied. The Riemann Problem is solved constructively. There are five kinds of Riemann solutions, in which four only contain different contact discontinuities and the other involves delta shock waves. Under suitable generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.
