Simple waves in quasilinear hyperbolic systems. II. Riemann invariants for the problem of simple wave interactions

1983 ◽  
Vol 24 (9) ◽  
pp. 2315-2328 ◽  
Author(s):  
Alfred M. Grundland ◽  
Roman Żelazny
Keyword(s):  
Hyperbolic Systems ◽  
Simple Wave ◽  
Riemann Invariants ◽  
Wave Interactions ◽  
Quasilinear Hyperbolic Systems ◽  
Simple Waves

Rotating Shallow-Water Models as Quasilinear Hyperbolic Systems, and Related Numerical Methods

2018 ◽  
Author(s):  
Vladimir Zeitlin
Keyword(s):  
Numerical Methods ◽  
Wave Breaking ◽  
Hyperbolic Systems ◽  
Simple Wave ◽  
Shear Instability ◽  
Lagrangian Description ◽  
Quasilinear Hyperbolic Systems ◽  
Hydraulic Theory ◽  
Shallow Water Models ◽  
Rotating Shallow Water

The chapter contains the mathematical background necessary to understand the properties of RSW models and numerical methods for their simulations. Mathematics of RSW model is presented by using their one-dimensional reductions, which are necessarily’one-and-a-half’ dimensional, due to rotation and include velocity in the second direction. Basic notions of quasi-linear hyperbolic systems are recalled. The notions of weak solutions, wave breaking, and shock formation are introduced and explained on the example of simple-wave equation. Lagrangian description of RSW is used to demonstrate that rotation does not prevent wave-breaking. Hydraulic theory and Rankine–Hugoniot jump conditions are formulated for RSW models. In the two-layer case it is shown that the system loses hyperbolicity in the presence of shear instability. Ideas of construction of well-balanced (i.e. maintaining equilibria) shock-resolving finite-volume numerical methods are explained and these methods are briefly presented, with illustrations on nonlinear evolution of equatorial waves.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

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