Stability of Entropy Solution of Conservation Law Systems on Flow Function and Relaxation Function

2021 ◽  
Vol 37 (4) ◽  
pp. 747-757
Author(s):  
Jin-bo Geng ◽  
Xu-dong Wang
Keyword(s):  
Conservation Law ◽  
Relaxation Function ◽  
Entropy Solution ◽  
Flow Function

scholarly journals Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions

2017 ◽  
Vol 9 (3) ◽  
pp. 515-542
Author(s):  
K. H. Karlsen ◽  
J. D. Towers
Keyword(s):  
Boundary Conditions ◽  
Conservation Law ◽  
Entropy Solution ◽  
Approximate Solutions ◽  
Convergence Result ◽  
Simple Modification ◽  
Monotone Schemes ◽  
Scalar Conservation Law ◽  
Flux Boundary Conditions ◽  
Scalar Conservation

AbstractWe consider a scalar conservation law with zero-flux boundary conditions imposed on the boundary of a rectangular multidimensional domain. We study monotone schemes applied to this problem. For the Godunov version of the scheme, we simply set the boundary flux equal to zero. For other monotone schemes, we additionally apply a simple modification to the numerical flux. We show that the approximate solutions produced by these schemes converge to the unique entropy solution, in the sense of [7], of the conservation law. Our convergence result relies on a BV bound on the approximate numerical solution. In addition, we show that a certain functional that is closely related to the total variation is nonincreasing from one time level to the next. We extend our scheme to handle degenerate convection-diffusion equations and for the one-dimensional case we prove convergence to the unique entropy solution.

A solution of the conservation law form of the Serre equations

2016 ◽  
Vol 57 ◽  
pp. 385
Author(s):  
Christopher Zoppou ◽  
Stephen Roberts ◽  
Jason Pitt
Keyword(s):  
Conservation Law

Relaxation function in regular and irregular regions of the Barbanis Hamiltonian

1990 ◽  
Vol 41 (6) ◽  
pp. 3070-3073
Author(s):  
B. Lauritzen ◽  
R. A. Broglia ◽  
M. Borromeo ◽  
W. E. Ormand
Keyword(s):  
Relaxation Function

scholarly journals New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mohammad M. Al-Gharabli ◽  
Adel M. Al-Mahdi ◽  
Salim A. Messaoudi
Keyword(s):  
Wide Class ◽  
Relaxation Function ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
General Assumption ◽  
General Decay ◽  
Source Terms ◽  
Relaxation Functions ◽  
Viscoelastic Equations ◽  
General Source

Abstract This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function $k_{i}$ k i , namely, $$\begin{aligned} k_{i}^{\prime }(t)\le -\xi _{i}(t) \Psi _{i} \bigl(k_{i}(t)\bigr),\quad i=1,2. \end{aligned}$$ k i ′ ( t ) ≤ − ξ i ( t ) Ψ i ( k i ( t ) ) , i = 1 , 2 . We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when $k_{i}(s) = s^{p}$ k i ( s ) = s p and p covers the full admissible range $[1, 2)$ [ 1 , 2 ) .

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