VISCOUS SHOCK PROFILES FOR 2 × 2 SYSTEMS OF HYPERBOLIC CONSERVATION LAWS WITH AN UMBILIC POINT

2009 ◽  
Vol 06 (03) ◽  
pp. 483-524
Author(s):  
FUMIOKI ASAKURA ◽  
MITSURU YAMAZAKI
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Propagation Speed ◽  
Umbilic Point ◽  
Hyperbolic Conservation ◽  
Reservoir Flow ◽  
Viscous Profile ◽  
Systems Of Conservation Laws ◽  
Shock Profiles ◽  
Shock Profile

This article analyzes the existence of viscous shock profiles joining two states satisfying the Rankine–Hugoniot condition that comes from hyperbolic 2 × 2 systems of conservation laws having quadratic flux functions with an isolated umbilic point: the point where the characteristic speeds coincide and the Jacobian matrix of the flux functions is diagonalizable. The systems studied in this note are particularly in Schaeffer and Shearer's cases I and II which are relevant to the three-phase Buckley–Leverett model for oil reservoir flow. It is shown that any compressive and overcompressive shocks have a viscous shock profile provided that there are no undercompressive shock with viscous profile having the same propagation speed. The idea of the proof is a generalization of the first theorem of Morse to noncompact level sets. It is also shown that there exists a shock satisfying the Liu–Oleĭnik condition but having no viscous shock profile. In this case, there is an undercompressive shock with viscous shock profile.

Linear stability of shock profiles for a quasilinear Benney system in ℝ2 × ℝ+

2020 ◽  
Vol 17 (04) ◽  
pp. 797-807
Author(s):  
João-Paulo Dias
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Short Wave ◽  
Wave Interactions ◽  
Hyperbolic Conservation ◽  
Linearized Stability ◽  
Shock Profiles ◽  
Scalar Conservation ◽  
Semilinear Schrödinger Equation ◽  
Shock Profile

Following Dias et al. [Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws, J. Differential Equations 251 (2007) 555–563], we study the linearized stability of a pair [Formula: see text], where [Formula: see text] is a shock profile for a family of quasilinear hyperbolic conservation laws in [Formula: see text] coupled with a semilinear Schrödinger equation.

scholarly journals Symmetries of conservation laws

2005 ◽  
Vol 77 (91) ◽  
pp. 29-51
Author(s):  
Sanja Konjik
Keyword(s):  
Shock Waves ◽  
Conservation Laws ◽  
Gas Dynamics ◽  
Hyperbolic Conservation Laws ◽  
Strong Association ◽  
Group Analysis ◽  
Symmetry Groups ◽  
Dynamics Model ◽  
Hyperbolic Conservation ◽  
Systems Of Conservation Laws

We apply techniques of symmetry group analysis in solving two systems of conservation laws: a model of two strictly hyperbolic conservation laws and a zero pressure gas dynamics model, which both have no global solution, but whose solution consists of singular shock waves. We show that these shock waves are solutions in the sense of 1-strong association. Also, we compute all project able symmetry groups and show that they are 1-strongly associated, hence transform existing solutions in the sense of 1-strong association into other solutions.

scholarly journals Nonoscillatory Central Schemes for Hyperbolic Systems of Conservation Laws in Three-Space Dimensions

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Andrew N. Guarendi ◽  
Abhilash J. Chandy
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Scaling Analysis ◽  
Hyperbolic Conservation ◽  
Central Schemes ◽  
Systems Of Conservation Laws ◽  
Ideal Magnetohydrodynamic ◽  
Three Space

We extend a family of high-resolution, semidiscrete central schemes for hyperbolic systems of conservation laws to three-space dimensions. Details of the schemes, their implementation, and properties are presented together with results from several prototypical applications of hyperbolic conservation laws including a nonlinear scalar equation, the Euler equations of gas dynamics, and the ideal magnetohydrodynamic equations. Parallel scaling analysis and grid-independent results including contours and isosurfaces of density and velocity and magnetic field vectors are shown in this study, confirming the ability of these types of solvers to approximate the solutions of hyperbolic equations efficiently and accurately.

scholarly journals A Weno-Tvd Implementation for Solving Some Problems of Hyperbolic Conservation Laws

2018 ◽  
Author(s):  
Jhon Alberto Polo Vásquez ◽  
Miguel Antonio Caro Candezano
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Numerical Solutions ◽  
Order Approximation ◽  
Computer Code ◽  
Tvd Scheme ◽  
Hyperbolic Conservation ◽  
Flux Limiter ◽  
Systems Of Conservation Laws ◽  
Fifth Order

This work deals with a numerical implementation of a fifth order CENTRAL WENO-TVD (\textit{Weighted Essentially Non-Oscillatory-Total Variation Dimimishing}) of Haschem (2006) scheme applied to the convective terms of some hyperbolic conservation laws problems, in a volume finite framework. The WENO-TVD scheme is used to solve the 1D advection and Burgers equations. For this case is implemented two different numerical fluxes: The Lax-Friedrichs and TVD fluxes. In the TVD fluxes the schemes applied are in flux-limiter form. The schemes implemented for this flux are: Van Albada-1 (van Albada et al.,1982), van Albada-2 (Kermani et al., 2003), van Leer (Hassanzadeh, 2009) and MINMOD (Hirsch, 2007). The WENO type schemes are characterized for their high order approximation, and do not produce spurious oscilations near discontinuities, shocks and higher gradients. A third order Runge-Kutta TVD for the temporal variable is used. Qualitative and quantitative comparison are presented. The numerical solutions are computed with an in-house computer code developed in MATLAB software. In future works, it will develope a paralelization of computer code for solving systems of conservation laws, e.g. Euler equations of gas dynamics.

AN ALTERNATING EVOLUTION APPROXIMATION TO SYSTEMS OF HYPERBOLIC CONSERVATION LAWS

2008 ◽  
Vol 05 (02) ◽  
pp. 421-447 ◽  
Author(s):  
HAILIANG LIU
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Entropy Solution ◽  
Approximate Solutions ◽  
Spatial Dimension ◽  
Approximation Formula ◽  
Method Error ◽  
Hyperbolic Conservation ◽  
Spatial Redistribution ◽  
Systems Of Conservation Laws

In this paper, we present an alternating evolution (AE) approximation [Formula: see text] to systems of hyperbolic conservation laws [Formula: see text] in arbitrary spatial dimension. We prove the convergence of the approximate solutions towards an entropy solution of scalar multi-D conservation laws, and the L1 contraction property for the approximate solution is established as well. It is also shown that such an approximation is extremely accurate in the sense that if initial data is prepared such that u0 = v0 = U0, then no method error is induced as time evolves, and the exact entropy solution is precisely captured. Furthermore, in the approximation system time evolution of one variable is associated with spatial redistribution in another variable. These features render such an approximation ideal to be used for construction of high resolution numerical schemes to solve hyperbolic conservation laws. The usual obstacles caused by jumps crossing computational cell interfaces are not felt when both u and v are sampled alternatively, and reconstructed independently. Herewith we discuss the designing principle for constructing AE schemes, with illustration of two preliminary schemes for systems of conservation laws in one dimension. Both l∞ monotonicity and the TVD (Total Variational Diminishing) property are established for these schemes when applied to the scalar laws.

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