scholarly journals Optimal jump set in hyperbolic conservation laws

2020 ◽  
Vol 17 (04) ◽  
pp. 765-784
Author(s):  
Shyam Sundar Ghoshal ◽  
Animesh Jana
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Entropy Solution ◽  
Hyperbolic Systems ◽  
Scalar Case ◽  
Qualitative Properties ◽  
Hyperbolic Conservation ◽  
Scalar Conservation Law ◽  
Scalar Conservation ◽  
Jump Set

We investigate qualitative properties of entropy solutions to hyperbolic conservation laws, and construct an entropy solution to a scalar conservation law for which the jump set is not closed, in particular, it is dense in a space-time domain. In a second part, we establish a similar result for hyperbolic systems. We give two different approaches for scalar conservation laws and hyperbolic systems in order to obtain these results. For the scalar case, the solutions are explicitly calculated.

scholarly journals Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

2019 ◽  
Vol 16 (03) ◽  
pp. 519-593
Author(s):  
L. Galimberti ◽  
K. H. Karlsen
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Generalized Solutions ◽  
Well Posedness ◽  
Scalar Conservation Laws ◽  
Stochastic Forcing ◽  
Rigidity Result ◽  
Hyperbolic Conservation ◽  
The Cauchy Problem ◽  
Scalar Conservation

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Itô) noise. The Cauchy problem defined on a Riemanian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result àla Perthame is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data ([Formula: see text] contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by Ben-Artzi and LeFloch [Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 24(6) (2007) 989–1008], who worked with Kružkov–DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle [Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259(4) (2010) 1014–1042].

LARGE TIME BEHAVIOR OF NUMERICAL SOLUTIONS OF SCALAR CONSERVATION LAWS

2006 ◽  
Vol 03 (04) ◽  
pp. 631-648
Author(s):  
FRÉDÉRIC LAGOUTIÈRE
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Large Time ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Hyperbolic Conservation ◽  
Periodic Initial Data ◽  
Scalar Conservation

We study the large time behavior of entropic approximate solutions to one-dimensional, hyperbolic conservation laws with periodic initial data. Under mild assumptions on the numerical scheme, we prove the asymptotic convergence of the discrete solutions to a time- and space-periodic solution.

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.

Basic structures of hyperbolic relaxation systems

2002 ◽  
Vol 132 (5) ◽  
pp. 1259-1274 ◽  
Author(s):  
Wen-An Yong
Keyword(s):  
Small Parameter ◽  
Conservation Laws ◽  
Structural Properties ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Systems ◽  
Source Terms ◽  
Hyperbolic Conservation ◽  
First Order ◽  
Relaxation Systems

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.

Basic structures of hyperbolic relaxation systems

2002 ◽  
Vol 132 (5) ◽  
pp. 1259-1274 ◽  
Author(s):  
Wen-An Yong
Keyword(s):  
Small Parameter ◽  
Conservation Laws ◽  
Structural Properties ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Systems ◽  
Source Terms ◽  
Hyperbolic Conservation ◽  
First Order ◽  
Relaxation Systems

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.

scholarly journals On the structure of BV entropy solutions for hyperbolic systems of balance laws with general flux function

2019 ◽  
Vol 16 (02) ◽  
pp. 333-378
Author(s):  
Fabio Ancona ◽  
Laura Caravenna ◽  
Andrea Marson
Keyword(s):  
Conservation Laws ◽  
Wave Front ◽  
Hyperbolic System ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Systems ◽  
Entropy Solutions ◽  
Balance Laws ◽  
Hyperbolic Conservation ◽  
Flux Function ◽  
Systems Of Balance Laws

The paper describes the qualitative structure of BV entropy solutions of a general strictly hyperbolic system of balance laws with characteristic field either piecewise genuinely nonlinear or linearly degenerate. In particular, we provide an accurate description of the local and global wave-front structure of a BV solution generated by a fractional step scheme combined with a wave-front tracking algorithm. This extends the corresponding results in [S. Bianchini and L. Yu, Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension, Comm. Partial Differential Equations 39(2) (2014) 244–273] for strictly hyperbolic system of conservation laws.

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.

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