scholarly journals Multidimensional Riemann problem with self-similar internal structure – part III – a multidimensional analogue of the HLLI Riemann solver for conservative hyperbolic systems

2017 ◽  
Vol 346 ◽  
pp. 25-48 ◽  
Author(s):  
Dinshaw S. Balsara ◽  
Boniface Nkonga
Keyword(s):  
Internal Structure ◽  
Riemann Problem ◽  
Hyperbolic Systems ◽  
Riemann Solver ◽  
Multidimensional Analogue ◽  
Self Similar

scholarly journals Coupling techniques for nonlinear hyperbolic equations. I Self-similar diffusion for thin interfaces

2011 ◽  
Vol 141 (5) ◽  
pp. 921-956 ◽  
Author(s):  
Benjamin Boutin ◽  
Frédéric Coquel ◽  
Philippe G. LeFloch
Keyword(s):  
Riemann Problem ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Resonance Effect ◽  
Wave Interaction ◽  
Technical Difficulty ◽  
Nonlinear Hyperbolic Equations ◽  
Global Hyperbolicity ◽  
Nonlinear Hyperbolic System ◽  
Self Similar

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial-value problem, and these solutions need to be supplemented with further admissibility conditions. This paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns.

THE SHOCK CURVE APPROACH TO THE RIEMANN PROBLEM FOR 2 × 2 HYPERBOLIC SYSTEMS OF CONSERVATION LAWS

2010 ◽  
Vol 07 (02) ◽  
pp. 339-364 ◽  
Author(s):  
HIROKI OHWA
Keyword(s):  
Conservation Laws ◽  
Riemann Problem ◽  
Space Variable ◽  
Hyperbolic Systems ◽  
Fréchet Derivative ◽  
Frechet Derivative ◽  
Systems Of Conservation Laws ◽  
Self Similar ◽  
Similar Solutions

We consider the Riemann problem for 2 × 2 hyperbolic systems of conservation laws in one space variable. Our main assumptions are that the product of non-diagonal elements within the Fréchet derivative (Jacobian) of the flux is positive, and that the system is genuinely nonlinear. The first assumption implies that the system is strictly hyperbolic, but we do not require a convexity-like condition such as the Smoller–Johnson condition. By using the shock curve approach, we show that those two assumptions are sufficient to establish the uniqueness of self-similar solutions satisfying the Lax entropy conditions at discontinuities.

scholarly journals A two-dimensional Riemann solver with self-similar sub-structure – Alternative formulation based on least squares projection

2016 ◽  
Vol 304 ◽  
pp. 138-161 ◽  
Author(s):  
Dinshaw S. Balsara ◽  
Jeaniffer Vides ◽  
Katharine Gurski ◽  
Boniface Nkonga ◽  
Michael Dumbser ◽  
...  
Keyword(s):  
Least Squares ◽  
Alternative Formulation ◽  
Riemann Solver ◽  
Two Dimensional ◽  
Self Similar

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.

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