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.

scholarly journals An exact Riemann-solver-based solution for regular shock refraction

2009 ◽  
Vol 627 ◽  
pp. 33-53 ◽  
Author(s):  
P. DELMONT ◽  
R. KEPPENS ◽  
B. VAN DER HOLST
Keyword(s):  
Single Point ◽  
Classical Problem ◽  
Wave Pattern ◽  
Riemann Solver ◽  
Contact Interface ◽  
Von Neumann ◽  
Density Discontinuity ◽  
Planar Shock ◽  
Shock Refraction ◽  
Exact Riemann Solver

We study the classical problem of planar shock refraction at an oblique density discontinuity, separating two gases at rest. When the shock impinges on the density discontinuity, it refracts, and in the hydrodynamical case three signals arise. Regular refraction means that these signals meet at a single point, called the triple point. After reflection from the top wall, the contact discontinuity becomes unstable due to local Kelvin–Helmholtz instability, causing the contact surface to roll up and develop the Richtmyer–Meshkov instability (RMI). We present an exact Riemann-solver-based solution strategy to describe the initial self-similar refraction phase, by which we can quantify the vorticity deposited on the contact interface. We investigate the effect of a perpendicular magnetic field and quantify how its addition increases the deposition of vorticity on the contact interface slightly under constant Atwood number. We predict wave-pattern transitions, in agreement with experiments, von Neumann shock refraction theory and numerical simulations performed with the grid-adaptive code AMRVAC. These simulations also describe the later phase of the RMI.

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.

1D Exact Elastic-Perfectly Plastic Solid Riemann Solver and Its Multi-Material Application

2017 ◽  
Vol 9 (3) ◽  
pp. 621-650 ◽  
Author(s):  
Si Gao ◽  
Tiegang Liu
Keyword(s):  
Riemann Problem ◽  
High Speed ◽  
Hyperbolic Conservation Laws ◽  
Riemann Solver ◽  
Hyperbolic Conservation ◽  
Perfectly Plastic ◽  
Impact Problems ◽  
Elastic Perfectly Plastic ◽  
Modified Ghost Fluid Method ◽  
Exact Riemann Solver

AbstractThe equation of state (EOS) plays a crucial role in hyperbolic conservation laws for the compressible fluid. Whereas, the solid constitutive model with elastic-plastic phase transition makes the analysis of the solid Riemann problem more difficult. In this paper, one-dimensional elastic-perfectly plastic solid Riemann problem is investigated and its exact Riemann solver is proposed. Different from previous works treating the elastic and plastic phases integrally, we resolve the elastic wave and plastic wave separately to understand the complicate nonlinear waves within the solid and then assemble them together to construct the exact Riemann solver for the elastic-perfectly plastic solid. After that, the exact solid Riemann solver is associated with the fluid Riemann solver to decouple the fluid-solid multi-material interaction. Numerical tests, including gas-solid, water-solid high-speed impact problems are simulated by utilizing the modified ghost fluid method (MGFM).

The development of a Riemann solver for the steady supersonic Euler equations

1994 ◽  
Vol 98 (979) ◽  
pp. 325-339 ◽  
Author(s):  
E. F. Toro ◽  
A. Chakraborty
Keyword(s):  
Euler Equations ◽  
Weighted Average ◽  
Godunov Method ◽  
Riemann Solver ◽  
Flux Method ◽  
First Order ◽  
Enhanced Resolution ◽  
Average Flux ◽  
Exact Riemann Solver ◽  
Order Weighted Average

Abstract An improved version (HLLC) of the Harten, Lax, van Leer Riemann solver (HLL) for the steady supersonic Euler equations is presented. Unlike the HLL, the HLLC version admits the presence of the slip line in the structure of the solution. This leads to enhanced resolution of computed slip lines by Godunov type methods. We assess the HLLC solver in the context of the first order Godunov method and the second order weighted average flux method (WAF). It is shown that the improvement embodied in the HLLC solver over the HLL solver is virtually equivalent to incorporating the exact Riemann solver.

scholarly journals Assessing the Ensemble Predictability of Precipitation Forecasts for the January 2015 and 2016 East Coast Winter Storms

2017 ◽  
Vol 32 (3) ◽  
pp. 1057-1078 ◽  
Author(s):  
Steven J. Greybush ◽  
Seth Saslo ◽  
Richard Grumm
Keyword(s):  
East Coast ◽  
Initial Conditions ◽  
Wave Pattern ◽  
Forecast Errors ◽  
Ensemble Forecasts ◽  
Winter Storms ◽  
Position Errors ◽  
Skill Scores ◽  
Forecast Lead Time ◽  
East Coast Winter Storms

Abstract The ensemble predictability of the January 2015 and 2016 East Coast winter storms is assessed, with model precipitation forecasts verified against observational datasets. Skill scores and reliability diagrams indicate that the large ensemble spread produced by operational forecasts was warranted given the actual forecast errors imposed by practical predictability limits. For the 2015 storm, uncertainties along the western edge’s sharp precipitation gradient are linked to position errors of the coastal low, which are traced to the positioning of the preceding 500-hPa wave pattern using the ensemble sensitivity technique. Predictability horizon diagrams indicate the forecast lead time in terms of initial detection, emergence of a signal, and convergence of solutions for an event. For the 2016 storm, the synoptic setup was detected at least 6 days in advance by global ensembles, whereas the predictability of mesoscale features is limited to hours. Convection-permitting WRF ensemble forecasts downscaled from the GEFS resolve mesoscale snowbands and demonstrate sensitivity to synoptic and mesoscale ensemble perturbations, as evidenced by changes in location and timing. Several perturbation techniques are compared, with stochastic techniques [the stochastic kinetic energy backscatter scheme (SKEBS) and stochastically perturbed parameterization tendency (SPPT)] and multiphysics configurations improving performance of both the ensemble mean and spread over the baseline initial conditions/boundary conditions (IC/BC) perturbation run. This study demonstrates the importance of ensembles and convective-allowing models for forecasting and decision support for east coast winter storms.

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 Hydrodynamic simulations of AGN jets: the impact of Riemann solvers and spatial reconstruction schemes on jet evolution

2020 ◽  
Vol 498 (3) ◽  
pp. 3870-3887
Author(s):  
G Musoke ◽  
A J Young ◽  
M Birkinshaw
Keyword(s):  
Active Galactic Nuclei ◽  
Numerical Algorithms ◽  
Galactic Nuclei ◽  
Riemann Solver ◽  
Third Order ◽  
Riemann Solvers ◽  
Hydrodynamic Simulations ◽  
Spatial Reconstruction ◽  
High Level ◽  
The Impact

ABSTRACT Numerical simulations play an essential role in helping us to understand the physical processes behind relativistic jets in active galactic nuclei. The large number of hydrodynamic codes available today enables a variety of different numerical algorithms to be utilized when conducting the simulations. Since many of the simulations presented in the literature use different combinations of algorithms it is important to quantify the differences in jet evolution that can arise due to the precise numerical schemes used. We conduct a series of simulations using the flash (magneto-)hydrodynamics code in which we vary the Riemann solver and spatial reconstruction schemes to determine their impact on the evolution and dynamics of the jets. For highly refined grids the variation in the simulation results introduced by the different combinations of spatial reconstruction scheme and Riemann solver is typically small. A high level of convergence is found for simulations using third-order spatial reconstruction with the Harten–Lax–Van-Leer with contact and Hybrid Riemann solvers.

scholarly journals Non-classical Riemann solvers with nucleation

2004 ◽  
Vol 134 (5) ◽  
pp. 961-984 ◽  
Author(s):  
P. G. LeFloch ◽  
M. Shearer
Keyword(s):  
Propagation Speed ◽  
Riemann Solver ◽  
Scalar Conservation Laws ◽  
Wave Interactions ◽  
Kinetic Relation ◽  
Riemann Solvers ◽  
Flux Function ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
Film Flows

We introduce a new non-classical Riemann solver for scalar conservation laws with concave–convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (under-compressive) non-classical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a non-classical one. We establish the existence of (non-classical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting–merging solutions, which are made of two large shocks and small bounded-variation perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin-film flows; they help explain numerical results observed for such flows.

scholarly journals Uniqueness of rarefaction waves in multidimensional compressible Euler system

2015 ◽  
Vol 12 (03) ◽  
pp. 489-499 ◽  
Author(s):  
Eduard Feireisl ◽  
Ondřej Kreml
Keyword(s):  
Shock Wave ◽  
Rarefaction Wave ◽  
Riemann Problem ◽  
Euler System ◽  
Entropy Solutions ◽  
Infinitely Many Solutions ◽  
Rarefaction Waves ◽  
Convex Integration ◽  
Wave Solutions ◽  
Compressible Euler

We show that 1D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.

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