Hyperbolic conservation laws with relaxation

Liu, Tai-Ping

doi:10.1007/bf01210707

Cited by 476 publications

(456 citation statements)

References 7 publications

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“…Additionally he conjectures a criterion for the non-appearance of spurious solutions in implicit split schemes for dissipative 2 x 2 systems, namely the commutability of the vanishing viscosity limit for viscous regularization of (1.7), and the limit of infinite stiffness. Liu (1987) and Chen et al (1992) show that for dissipative 2 x 2 systems this commutability holds.…”

Section: Description Of the Algorithm And Numerical Resultsmentioning

confidence: 99%

Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in One Space Dimension

Papalexandris¹,

Leonard²,

Dimotakis³

1997

Journal of Computational Physics

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Abstract.The present work is concerned with the extension of the theory of characteristics to conservation laws with source terms in one space dimension, such as the Euler equations for reacting flows. New spacetime curves are introduced on which the equations decouple to the characteristic set of O.D.E's for the corresponding homogeneous laws, thus allowing the introduction of functions analogous to the Riemann Invariants. The geometry of these curves depends on the spatial gradients for the solution. This particular decomposition can be used in the design of efficient unsplit algorithms for the numerical integration of the equations. As a first step, these ideas are implemented for the case of a scalar conservation law with a nonlinear source term. The resulting algorithm belongs to the class of MUSCL-type, shock-capturing schemes. Its accuracy and robustness are checked through a series of tests. The aspect of the stiffness of the source term is also studied. Then, the algorithm is generalized for a system of hyperbolic equations, namely the Euler equations for reacting flows. An extensive numerical study of unstable detonations is performed.

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Section: Description Of the Algorithm And Numerical Resultsmentioning

confidence: 99%

Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in One Space Dimension

Papalexandris¹,

Leonard²,

Dimotakis³

1997

Journal of Computational Physics

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“…it can be shown (Liu, 1987, The interpretation of the development of a traveling wave is this: Dispersion or adsorption kinetics tend to produce a smooth transition between c + and c -, whereas the nonlinear dependence of the adsorbed concentration C* on c counteracts dispersion or kinetics and tries to produce a steep profile. The traveling wave is the stationary compromise.…”

Section: Appendices A1 Relative Influence Of Diffusion/dispersion Anmentioning

confidence: 99%

“…ε is the corresponding relaxation time due to adsorption kinetics. Liu (1987) presented a solution for (A 2. 1), (A 2.…”

Section: Appendices A1 Relative Influence Of Diffusion/dispersion Anmentioning

confidence: 99%

Advective Transport of Interacting Solutes: The Chromatographic Model

Gruber¹

1996

Sediments and Toxic Substances

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Species on porous media surfaces exposed to aqueous solutions can still not yet be determined unambiguously. The influence of these ambiguities on contaminant transport needs to be understood. Here methods of multicomponent chromatography are presented that allow to visualize the effects of the ambiguities. The Riemann problem is solved to a large extent based on the mathematics of non-linear hyperbolic differential equations, before the problem is handed over to the computer. Since thereby the computational efforts are kept compatible with the corresponding limitations in a chemical laboratory, this method has been used for decades in chemical engineering. The method is illustrated for various multicomponent isotherms, e.g. the one underlying surface complexation models as implemented in the MINEQL family of programs.

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“…The limiting process → 0 in systems in the form (55) was extensively analysed by Liu [24] and Chen et al [4], with a particular focus on the relationship between stability and wave propagation. It is of high interest to obtain good numerical methods for systems in the form (55) when the relaxation source term is stiff; i.e.…”

Section: Hyperbolic Relaxation Systemsmentioning

confidence: 99%

An exponential time-differencing method for monotonic relaxation systems

Aursand

Evje

Flåtten

et al. 2014

Applied Numerical Mathematics

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Abstract. We present first and second-order accurate exponential time differencing methods for a special class of stiff ODEs, denoted as monotonic relaxation ODEs. Some desirable accuracy and robustness properties of our methods are established. In particular, we prove a strong form of stability denoted as monotonic asymptotic stability, guaranteeing that no overshoots of the equilibrium value are possible. This is motivated by the desire to avoid spurious unphysical values that could crash a large simulation.We present a simple numerical example, demonstrating the potential for increased accuracy and robustness compared to established Runge-Kutta and exponential methods. Through operator splitting, an application to granulargas flow is provided.

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Hyperbolic conservation laws with relaxation

Cited by 476 publications

References 7 publications

Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in One Space Dimension

Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in One Space Dimension

Advective Transport of Interacting Solutes: The Chromatographic Model

An exponential time-differencing method for monotonic relaxation systems

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