scholarly journals AQ-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system

2001 ◽  
Vol 35 (1) ◽  
pp. 107-127 ◽  
Author(s):  
Manuel Castro ◽  
Jorge Macías ◽  
Carlos Parés
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Source Term ◽  
Water System ◽  
Shallow Water System

On the Shallow Water Equations

2017 ◽  
Vol 72 (9) ◽  
pp. 873-879 ◽  
Author(s):  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water System ◽  
Diagonal Form ◽  
Riemann Invariants ◽  
Interesting Application ◽  
Nonlinear Conservation Laws ◽  
Shallow Water System ◽  
Elementary Waves

AbstractWe studied the shallow water equations of nonlinear conservation laws. First we studied the parametrisation of nonlinear elementary waves and hence we present the solution to the Riemann problem. We also prove the uniqueness of the Riemann solution. The Riemann invariants are formulated. Moreover we give an interesting application of the Riemann invariants. We present the shallow water system in a diagonal form, which admits the existence of a global smooth solution for these equations. The other application is to introduce new conservation laws for the shallow water equations.

scholarly journals Viscoelastic flows of Maxwell fluids with conservation laws

2020 ◽  
Author(s):  
Sebastien Boyaval
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Compressible Flows ◽  
Causal Model ◽  
Initial Conditions ◽  
Water System ◽  
Viscoelastic Flows ◽  
Viscous Effects ◽  
Additional Material ◽  
Shallow Water System

We consider multi-dimensional extensions of Maxwell’s seminal rheological equation for 1D viscoelastic flows. We aim at a causal model for compressible flows, defined by semi-group solutions given initial conditions, and such that perturbations propagates at finite speed. We propose a symmetric hyperbolic system of conservation laws that contains the Upper-Convected Maxwell (UCM) equation as causal model. The system is an extension of polyconvex elastodynamics, with an additional material metric variable that relaxes to model viscous effects. Interestingly, the framework could also cover other rheological equations, depending on the chosen relaxation limit for the material metric variable. We propose to apply the new system to incompressible free-surface gravity flows in the shallow-water regime, when causality is important. The system reduces to a viscoelastic extension of Saint-Venant 2D shallow-water system that is symmetric-hyperbolic and that encompasses our previous viscoelastic extensions of Saint-Venant proposed with F. Bouchut.

WELL-BALANCED NUMERICAL SCHEMES BASED ON A GENERALIZED HYDROSTATIC RECONSTRUCTION TECHNIQUE

2007 ◽  
Vol 17 (12) ◽  
pp. 2055-2113 ◽  
Author(s):  
MANUEL J. CASTRO ◽  
ALBERTO PARDO MILANÉS ◽  
CARLOS PARÉS
Keyword(s):  
Shallow Water ◽  
Numerical Scheme ◽  
Source Term ◽  
Hyperbolic Systems ◽  
Water System ◽  
The Other ◽  
Reconstruction Technique ◽  
Numerical Schemes ◽  
Shallow Water System ◽  
The One

The goal of this paper is to generalize the hydrostatic reconstruction technique introduced in Ref. 2 for the shallow water system to more general hyperbolic systems with source term. The key idea is to interpret the numerical scheme obtained with this technique as a path-conservative method, as defined in Ref. 35. This generalization allows us, on the one hand, to construct well-balanced numerical schemes for new problems, as the two-layer shallow water system. On the other hand, we construct numerical schemes for the shallow water system with better well-balanced properties. In particular we obtain a Roe method which solves exactly every stationary solution, and not only those corresponding to water at rest.

Global well-posedness for the viscous shallow water system with Korteweg type

2017 ◽  
Vol 97 (16) ◽  
pp. 2865-2879
Author(s):  
Jinlu Li ◽  
Zhaoyang Yin
Keyword(s):  
Shallow Water ◽  
Water System ◽  
Well Posedness ◽  
Shallow Water System

Delta Shock Waves in the Shallow Water System

2017 ◽  
Vol 30 (3) ◽  
pp. 1187-1198 ◽  
Author(s):  
C. O. R. Sarrico ◽  
A. Paiva
Keyword(s):  
Shock Waves ◽  
Shallow Water ◽  
Water System ◽  
Delta Shock ◽  
Shallow Water System ◽  
Delta Shock Waves
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