scholarly journals Riemann problem for rate‐type materials with nonconstant initial conditions

2021 ◽  
Author(s):  
R. Radha ◽  
Vishnu Dutt Sharma ◽  
Akshay Kumar
Keyword(s):  
Riemann Problem ◽  
Initial Conditions ◽  
Rate Type

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.

Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics

2003 ◽  
Vol 69 (3) ◽  
pp. 253-276 ◽  
Author(s):  
M. TORRILHON
Keyword(s):  
Riemann Problem ◽  
Initial Conditions ◽  
Regular Waves ◽  
Regular Solutions ◽  
Riemann Solutions ◽  
Flux Function ◽  
Ideal Mhd ◽  
Ideal Magnetohydrodynamics ◽  
Mhd System ◽  
Hugoniot Curves

The equations of ideal magnetohydrodynamics (MHD) form a non-strict hyperbolic system with a non-convex flux function and admit non-regular, so-called intermediate shocks. The presence of non-regular waves in the MHD system causes the Riemann problem to be not unique in some cases. This paper investigates the uniqueness of Riemann solutions of ideal MHD. To determine uniqueness conditions we discuss the correspondence of non-regular solutions and non-uniqueness. Additionally the structure of the Hugoniot curves and its non-regular behaviour are demonstrated. It follows that the degree of freedom for solving a Riemann problem is reduced in the case of a non-regular solution. From this, we can deduce uniqueness conditions depending on the initial conditions of an MHD Riemann problem. The results also allow one to construct non-unique solutions. We give an example for the case of non-planar initial conditions.

scholarly journals Multiple Solutions for the Riemann Problem in the Porous Shallow Water Equations

10.29007/31n4 ◽  
2018 ◽  
Author(s):  
Luca Cozzolino ◽  
Raffaele Castaldo ◽  
Luigi Cimorelli ◽  
Renata Della Morte ◽  
Veronica Pepe ◽  
...  
Keyword(s):  
Shallow Water ◽  
Riemann Problem ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Numerical Models ◽  
Control Volume ◽  
Fluid Phase ◽  
Problem Solution ◽  
The One ◽  
Solid Obstacles

The Porous Shallow water Equations are widely used in the context of urban flooding simulation. In these equations, the solid obstacles are implicitly taken into account by averaging the classic Shallow water Equations on a control volume containing the fluid phase and the obstacles. Numerical models for the approximate solution of these equations are usually based on the approximate calculation of the Riemann fluxes at the interface between cells. In the present paper, it is presented the exact solution of the one-dimensional Riemann problem over the dry bed, and it is shown that the solution always exists, but there are initial conditions for which it is not unique. The non-uniqueness of the Riemann problem solution opens interesting questions about which is the physically congruent wave configuration in the case of solution multiplicity.

Sebuah Generalisasi Baru Barisan Fibonacci-Lucas

2019 ◽  
Vol 2 (2) ◽  
Author(s):  
Musraini M Musraini M ◽  
Rustam Efendi ◽  
Rolan Pane ◽  
Endang Lily
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Lucas Sequence ◽  
Binet’S Formula

Barisan Fibonacci dan Lucas telah digeneralisasi dalam banyak cara, beberapa dengan mempertahankan kondisi awal, dan lainnya dengan mempertahankan relasi rekurensi. Makalah ini menyajikan sebuah generalisasi baru barisan Fibonacci-Lucas yang didefinisikan oleh relasi rekurensi B_n=B_(n-1)+B_(n-2),n≥2 , B_0=2b,B_1=s dengan b dan s bilangan bulat  tak negatif. Selanjutnya, beberapa identitas dihasilkan dan diturunkan menggunakan formula Binet dan metode sederhana lainnya. Juga dibahas beberapa identitas dalam bentuk determinan.   The Fibonacci and Lucas sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this paper, a new generalization of Fibonacci-Lucas sequence is introduced and defined by the recurrence relation B_n=B_(n-1)+B_(n-2),n≥2, with ,  B_0=2b,B_1=s                          where b and s are non negative integers. Further, some identities are generated and derived by Binet’s formula and other simple methods. Also some determinant identities are discussed.

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