Reduction of balance laws to conservation laws by means of equivalence transformations

2013 ◽  
Vol 54 (4) ◽  
pp. 041506 ◽  
Author(s):  
F. Oliveri ◽  
M. P. Speciale
Keyword(s):  
Conservation Laws ◽  
Balance Laws ◽  
Equivalence Transformations

Configurational balance laws for incompatibility in stress space

2007 ◽  
Vol 463 (2081) ◽  
pp. 1379-1392 ◽  
Author(s):  
Xanthippi Markenscoff ◽  
Anurag Gupta
Keyword(s):  
Conservation Laws ◽  
Thermal Stresses ◽  
Stress Gradient ◽  
Positive Definite ◽  
Balance Laws ◽  
Source Terms ◽  
Stress Space ◽  
Lagrange Equations ◽  
Compatibility Conditions ◽  
Surface Data

Conservation laws have been recently obtained by requiring that a positive definite functional of the stress gradient (the Euler–Lagrange equations of which are the Beltrami–Michell compatibility conditions) be invariant under certain transformations. Here these laws are extended to include body forces, thermal stresses and Kröner's incompatibility tensor as source terms in the configurational balance laws, which allows for the incompatibility in the volume to be measured from surface data. An example is presented.

Energy Release Rates and Related Balance Laws in Linear Elastic Defect Mechanics

1987 ◽  
Vol 54 (2) ◽  
pp. 388-392 ◽  
Author(s):  
J. W. Eischen ◽  
G. Herrmann
Keyword(s):  
Conservation Laws ◽  
Energy Release ◽  
Thermal Effects ◽  
Balance Laws ◽  
Release Rates ◽  
Direct Derivation ◽  
Linear Elastic ◽  
Linear Dynamic ◽  
Dynamic Elasticity ◽  
Energy Release Rates

A simple and direct derivation of certain balance (or conservation) laws for linear dynamic elasticity is presented including nonhomogeneities, thermal effects, anisotropy, and body forces. Additionally, the connection between the balance laws and energy release rates for defect motions is established.

scholarly journals Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws

2018 ◽  
Vol 13 (1) ◽  
pp. 191-200 ◽  
Author(s):  
Eduardo Abreu ◽  
◽  
John Perez ◽  
Arthur Santo ◽  
◽  
...  
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Approximation Methods ◽  
Balance Laws ◽  
Hyperbolic Conservation

scholarly journals A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions

2003 ◽  
Vol 24 (3) ◽  
pp. 955-978 ◽  
Author(s):  
Derek S. Bale ◽  
Randall J. LeVeque ◽  
Sorin Mitran ◽  
James A. Rossmanith
Keyword(s):  
Wave Propagation ◽  
Conservation Laws ◽  
Balance Laws ◽  
Wave Propagation Method ◽  
Spatially Varying

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 Stability and convergence of relaxation schemes to hyperbolic balance laws via a wave operator

2015 ◽  
Vol 12 (01) ◽  
pp. 189-219
Author(s):  
Alexey Miroshnikov ◽  
Konstantina Trivisa
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Stability And Convergence ◽  
Balance Laws ◽  
Source Terms ◽  
Hyperbolic Conservation ◽  
Hyperbolic Balance Laws ◽  
Relaxation Schemes ◽  
Systems Of Balance Laws ◽  
Weak Dissipation

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505–527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357–1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277] for systems of hyperbolic conservation laws without source terms.

scholarly journals Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Feng Zhang ◽  
Yuru Hu ◽  
Xiangpeng Xin
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Optimal System ◽  
Differential Invariants ◽  
Group Method ◽  
Equivalence Transformations ◽  
3 Dimensional ◽  
Dimensional Variable

In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ( 1 + 1 )-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.

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