scholarly journals Shadow wave solutions for a scalar two-flux conservation law with Rankine–Hugoniot deficit

2021 ◽  
Vol 18 (03) ◽  
pp. 539-556
Author(s):  
Tanja Krunić ◽  
Marko Nedeljkov
Keyword(s):  
Shock Wave ◽  
Weak Solution ◽  
Hyperbolic Conservation Laws ◽  
Jump Condition ◽  
Delta Function ◽  
Piecewise Constant ◽  
Hyperbolic Conservation ◽  
Unbounded Solutions ◽  
Wave Solutions ◽  
Flux Conservation

This paper deals with hyperbolic conservation laws exhibiting a flux discontinuity at the origin and which does not admit a weak solution satisfying the Rankine–Hugoniot jump condition. We therefore seek unbounded solutions in the form of shadow waves supported by at the origin. The shadow waves are defined as nets of piecewise constant functions approximating a shock wave to which we add a delta function and possibly another unbounded part.

scholarly journals Convergence of second-order, entropy stable methods for multi-dimensional conservation laws

2020 ◽  
Vol 54 (4) ◽  
pp. 1415-1428
Author(s):  
Neelabja Chatterjee ◽  
Ulrik Skre Fjordholm
Keyword(s):  
Weak Solution ◽  
Numerical Methods ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Compensated Compactness ◽  
Entropy Stability ◽  
Entropy Condition ◽  
Hyperbolic Conservation ◽  
Convergence Results ◽  
Entropy Stable

High-order accurate, entropy stable numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space dimensions. In this paper we show how the entropy stability of one such method, which is semi-discrete in time, yields a (weak) bound on oscillations. Under the assumption of L∞-boundedness of the approximations we use compensated compactness to prove convergence to a weak solution satisfying at least one entropy condition.

Finite volume methods for hyperbolic conservation laws

Acta Numerica ◽  
2007 ◽  
Vol 16 ◽  
pp. 155-238 ◽  
Author(s):  
K. W. Morton ◽  
T. Sonar
Keyword(s):  
Finite Volume ◽  
Hyperbolic Conservation Laws ◽  
Equation System ◽  
Finite Volume Methods ◽  
Optimal Recovery ◽  
Theory And Practice ◽  
Piecewise Constant ◽  
Hyperbolic Conservation ◽  
Discrete Equations ◽  
Finite Difference Equations

Finite volume methods apply directly to the conservation law form of a differential equation system; and they commonly yield cell average approximations to the unknowns rather than point values. The discrete equations that they generate on a regular mesh look rather like finite difference equations; but they are really much closer to finite element methods, sharing with them a natural formulation on unstructured meshes. The typical projection onto a piecewise constant trial space leads naturally into the theory of optimal recovery to achieve higher than first-order accuracy. They have dominated aerodynamics computation for over forty years, but they have never before been the subject of an Acta Numerica article. We shall therefore survey their early formulations before describing powerful developments in both their theory and practice that have taken place in the last few years.

scholarly journals High order explicit local time stepping methods for hyperbolic conservation laws

2020 ◽  
Vol 89 (324) ◽  
pp. 1807-1842
Author(s):  
Thi-Thao-Phuong Hoang ◽  
Lili Ju ◽  
Wei Leng ◽  
Zhu Wang
Keyword(s):  
Conservation Laws ◽  
Local Time ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Hyperbolic Conservation ◽  
Time Stepping ◽  
Local Time Stepping

scholarly journals Nematic Dispersive Shock Waves from Nonlocal to Local

2021 ◽  
Vol 11 (11) ◽  
pp. 4736
Author(s):  
Saleh Baqer ◽  
Dimitrios J. Frantzeskakis ◽  
Theodoros P. Horikis ◽  
Côme Houdeville ◽  
Timothy R. Marchant ◽  
...  
Keyword(s):  
Shock Wave ◽  
Liquid Crystals ◽  
Shock Waves ◽  
Numerical Solutions ◽  
Wave Structure ◽  
Beam Power ◽  
Input Beam ◽  
Shock Wave Structure ◽  
Wave Solutions ◽  
Modulation Theory

The structure of optical dispersive shock waves in nematic liquid crystals is investigated as the power of the optical beam is varied, with six regimes identified, which complements previous work pertinent to low power beams only. It is found that the dispersive shock wave structure depends critically on the input beam power. In addition, it is known that nematic dispersive shock waves are resonant and the structure of this resonance is also critically dependent on the beam power. Whitham modulation theory is used to find solutions for the six regimes with the existence intervals for each identified. These dispersive shock wave solutions are compared with full numerical solutions of the nematic equations, and excellent agreement is found.

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