scholarly journals Riemann problem for rate-type materials with non-constant initial conditions

2021 ◽  
Author(s):  
R. Radha ◽  
Vishnu Dutt Sharma ◽  
Akshay Kumar
Keyword(s):  
Initial Data ◽  
Exact Solutions ◽  
Riemann Problem ◽  
Initial Conditions ◽  
Differential Invariants ◽  
Quasilinear Hyperbolic System ◽  
Rarefaction Waves ◽  
Genuine Nonlinearity ◽  
Rate Type ◽  
Complete Characterisation

In this paper, using the compatible theory of differential invariants, a class of exact solutions is obtained for nonhomogeneous quasilinear hyperbolic system of partial differential equations (PDEs) describing rate type materials; these solutions exhibit genuine nonlinearity that leads to the formation of discontinuities such as shocks and rarefaction waves. For certain nonconstant and smooth initial data, the solution to the Riemann problem is presented providing a complete characterisation of the solutions.

scholarly journals An improved exact Riemann solver for relativistic hydrodynamics

2001 ◽  
Vol 449 ◽  
pp. 395-411 ◽  
Author(s):  
LUCIANO REZZOLLA ◽  
OLINDO ZANOTTI
Keyword(s):  
Riemann Problem ◽  
Initial Conditions ◽  
Wave Pattern ◽  
Initial Guess ◽  
Relativistic Velocity ◽  
Riemann Solver ◽  
Rarefaction Waves ◽  
Riemann Solvers ◽  
Velocity Jump ◽  
Exact Riemann Solver

A Riemann problem with prescribed initial conditions will produce one of three possible wave patterns corresponding to the propagation of the different discontinuities that will be produced once the system is allowed to relax. In general, when solving the Riemann problem numerically, the determination of the specific wave pattern produced is obtained through some initial guess which can be successively discarded or improved. We here discuss a new procedure, suitable for implementation in an exact Riemann solver in one dimension, which removes the initial ambiguity in the wave pattern. In particular we focus our attention on the relativistic velocity jump between the two initial states and use this to determine, through some analytic conditions, the wave pattern produced by the decay of the initial discontinuity. The exact Riemann problem is then solved by means of calculating the root of a nonlinear equation. Interestingly, in the case of two rarefaction waves, this root can even be found analytically. Our procedure is straightforward to implement numerically and improves the efficiency of numerical codes based on exact Riemann solvers.

STABILITY OF RAREFACTION WAVES AND VACUUM STATES FOR THE MULTIDIMENSIONAL EULER EQUATIONS

2007 ◽  
Vol 04 (01) ◽  
pp. 105-122 ◽  
Author(s):  
GUI-QIANG CHEN ◽  
JUN CHEN
Keyword(s):  
Initial Data ◽  
Large Class ◽  
Euler Equations ◽  
Riemann Problem ◽  
Global Attractors ◽  
Compressible Fluids ◽  
Entropy Solutions ◽  
Rarefaction Waves ◽  
One Dimensional ◽  
Vacuum States

We are interested in properties of the multidimensional Euler equations for compressible fluids. Rarefaction waves are the unique solutions that may contain vacuum states in later time, in the context of one-dimensional Riemann problem, even when the Riemann initial data are away from the vacuum. For the multidimensional Euler equations describing isentropic or adiabatic fluids, we prove that plane rarefaction waves and vacuum states are stable within a large class of entropy solutions that may contain vacuum states. Rarefaction waves and vacuum states are also shown to be global attractors of entropy solutions in L∞, provided initial data are L∞ ∩ L1 perturbations of Riemann initial data. Our analysis applies to entropy solutions with arbitrarily large oscillation, and no bounded variation regularity is required.

scholarly journals Hamiltonian charges in the asymptotically de Sitter spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Maciej Kolanowski ◽  
Jerzy Lewandowski
Keyword(s):  
Phase Space ◽  
Initial Data ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Physical Meaning ◽  
De Sitter ◽  
Killing Vectors ◽  
Asymptotically Flat ◽  
Angular Momenta ◽  
The One

Abstract We generalize a notion of ‘conserved’ charges given by Wald and Zoupas to the asymptotically de Sitter spacetimes. Surprisingly, our construction is less ambiguous than the one encountered in the asymptotically flat context. An expansion around exact solutions possessing Killing vectors provides their physical meaning. In particular, we discuss a question of how to define energy and angular momenta of gravitational waves propagating on Kottler and Carter backgrounds. We show that obtained expressions have a correct limit as Λ → 0. We also comment on the relation between this approach and the one based on the canonical phase space of initial data at ℐ+.

scholarly journals Method of Constructing a Line Secretive Exit the Aircraft at a Given Point of the Detection Area Surveillance Radar Based on the Spectrum Analysis of the Doppler Frequency of the Received Signal

2021 ◽  
pp. 281-291
Author(s):  
Valeriy V. Zamaraev ◽  
A.S. Kutuzov ◽  
Igor V. Lyutikov ◽  
Dmitry V. Malcev
Keyword(s):  
Initial Data ◽  
Spectrum Analysis ◽  
Initial Conditions ◽  
Doppler Frequency ◽  
Subject Area ◽  
Theory And Practice ◽  
Control Signals ◽  
The Subject ◽  
Area Surveillance

The article noted the contradictions in the theory and practice of the subject area of the study confirming the urgency of the task of constructing the trajectory of a hidden exit the aircraft at a given point of the detection area surveillance radar based on the spectrum analysis of the Doppler frequency of the received signal, the proposed initial data (initial conditions), describes the scientific and methodological apparatus of the embodiment of the method for generating control signals to implement the synthesized method, which allows to increase the effectiveness of air and space attack

scholarly journals Dependence on the Initial Data for the Continuous Thermostatted Framework

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 602
Author(s):  
Bruno Carbonaro ◽  
Marco Menale
Keyword(s):  
Complex System ◽  
Initial Data ◽  
External Force ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Complete Information ◽  
Complete Picture ◽  
Order Moment ◽  
Lack Of Information ◽  
Activity Variable

The paper deals with the problem of continuous dependence on initial data of solutions to the equation describing the evolution of a complex system in the presence of an external force acting on the system and of a thermostat, simply identified with the condition that the second order moment of the activity variable (see Section 1) is a constant. We are able to prove that these solutions are stable with respect to the initial conditions in the Hadamard’s sense. In this connection, two remarks spontaneously arise and must be carefully considered: first, one could complain the lack of information about the “distance” between solutions at any time t ∈ [ 0 , + ∞ ) ; next, one cannot expect any more complete information without taking into account the possible distribution of the transition probabiliy densities and the interaction rates (see Section 1 again). This work must be viewed as a first step of a research which will require many more steps to give a sufficiently complete picture of the relations between solutions (see Section 5).

A Second-Order Cell-Centered Lagrangian Method for Two-Dimensional Elastic-Plastic Flows

2017 ◽  
Vol 22 (5) ◽  
pp. 1224-1257 ◽  
Author(s):  
Jun-Bo Cheng ◽  
Yueling Jia ◽  
Song Jiang ◽  
Eleuterio F. Toro ◽  
Ming Yu
Keyword(s):  
Constitutive Model ◽  
Riemann Problem ◽  
Wave Structure ◽  
Solution Procedure ◽  
Second Order ◽  
Rarefaction Waves ◽  
Elastic Plastic ◽  
Yielding Condition ◽  
Nested Iterations ◽  
Von Mises

AbstractFor 2D elastic-plastic flows with the hypo-elastic constitutive model and von Mises’ yielding condition, the non-conservative character of the hypo-elastic constitutive model and the von Mises’ yielding condition make the construction of the solution to the Riemann problem a challenging task. In this paper, we first analyze the wave structure of the Riemann problem and develop accordingly aFour-Rarefaction wave approximateRiemannSolver withElastic waves (FRRSE). In the construction of FRRSE one needs to use an iterative method. A direct iteration procedure for four variables is complex and computationally expensive. In order to simplify the solution procedure we develop an iteration based on two nested iterations upon two variables, and our iteration method is simple in implementation and efficient. Based on FRRSE as a building block, we propose a 2nd-order cell-centered Lagrangian numerical scheme. Numerical results with smooth solutions show that the scheme is of second-order accuracy. Moreover, a number of numerical experiments with shock and rarefaction waves demonstrate the scheme is essentially non-oscillatory and appears to be convergent. For shock waves the present scheme has comparable accuracy to that of the scheme developed by Maire et al., while it is more accurate in resolving rarefaction waves.

scholarly journals Construction of the Global Solutions to the Perturbed Riemann Problem for the Leroux System

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Pengpeng Ji ◽  
Chun Shen
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Phase Plane ◽  
Wave Interaction ◽  
Global Solutions ◽  
Geometrical Structures ◽  
Small Perturbations ◽  
Riemann Solutions ◽  
Piecewise Constant ◽  
Rarefaction Curves

The global solutions of the perturbed Riemann problem for the Leroux system are constructed explicitly under the suitable assumptions when the initial data are taken to be three piecewise constant states. The wave interaction problems are widely investigated during the process of constructing global solutions with the help of the geometrical structures of the shock and rarefaction curves in the phase plane. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.

scholarly journals Regularization of the Shock Wave Solution to the Riemann Problem for the Relativistic Burgers Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Ting Zhang ◽  
Chun Shen
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Wave Solution ◽  
Burgers Equation ◽  
Explicit Computation ◽  
Convective Term ◽  
Regularization Technique ◽  
Shock Wave Solution

The regularization of the shock wave solution to the Riemann problem for the relativistic Burgers equation is considered. For Riemann initial data consisting of a single decreasing jump, we find that the regularization of nonlinear convective term cannot capture the correct shock wave solution. In order to overcome it, we consider a new regularization technique called the observable divergence method introduced by Mohseni and discover that it can capture the correct shock wave solution. In addition, we take the Helmholtz filter for the fully explicit computation.

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