Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Monographs and Surveys in Pure and Applied Mathematics, Vol. 118

Kulikovskii, A.; Pogorelov, N. V.; Semenov, A. Yu.; Piechór, K.

doi:10.1115/1.1470672

Cited by 27 publications

(18 citation statements)

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“…Up to notation, system (2.2) coincides with the system describing nonlinear electromagnetic waves in weakly anisotropic magnetic solids [3], [5].…”

Section: Discontinuity Structurementioning

confidence: 99%

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

Chugainova

2006

Theor Math Phys

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In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a "spreading" discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity.We consider specific features of the behavior of nonlinear quasitransverse waves in composite elastic media where the long-wave perturbations are described by nonlinear hyperbolic equations expressing the conservation laws. Discontinuities can appear in the solutions of these equations in their evolution process. Dispersion and dissipation are essential in smaller-scale phenomena. To describe the processes in narrow regions that are replaced with discontinuities for large scales, we use a more complete system of equations taking the large-and small-scale phenomena into account. The discontinuities associated with a continuous traveling-wave-type solution of the complete system of equations (the solution of the problem of finding the discontinuity structure) are usually said to be admissible. The influence of dispersion and dissipation results in the structure of the set of admissible discontinuities becoming rather complicated. We assume that a hyperbolic system of equations with the corresponding system of relations for the discontinuities and with the additional requirement that the discontinuities belong to the set of admissible discontinuities is taken as a model of large-scale phenomena. Then, as was shown in [1]-[3] for nonlinear waves in rods and magnetic solids as well as for a certain model of composite media, the solutions of self-similar problems for some distinguished domains of initial parameters can be nonunique. The number of solutions in these domains depends on the dispersion inside the discontinuity structure (for a constant viscosity) and increases without bound as the dispersion increases. In this paper, we numerically analyze self-similar asymptotic approximations in the nonuniqueness domains as a result of the evolution of nonstationary solutions of the complete system of partial differential equations. A model of large-scale phenomenaIf the medium anisotropy is small, then weakly nonlinear quasitransverse waves propagating in the positive direction of the x axis can be described by a simplified system of equations obtained from the

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“…Up to notation, system (2.2) coincides with the system describing nonlinear electromagnetic waves in weakly anisotropic magnetic solids [3], [5].…”

Section: Discontinuity Structurementioning

confidence: 99%

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

Chugainova

2006

Theor Math Phys

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“…We apply the Riemann solver developed by Prokopov [31] based on the exact solution of the nonheating PDE system (for description of this Riemann solver see also [36]). The "exact solution" means that we determine accurately the wave pattern emerging after left-side and right-side gases (each with constant parameters) begin to interact.…”

Section: Riemann Problem On the Moving Meshmentioning

confidence: 99%

“…piston. The fact is that we perform calculations up to time t≈62 and, therefore, with the steady ZND's mode values D o =7.459, u bnt =4.704, c bnt =4.0 and initial L dom (0)=5L 1/2 via (36) one estimates the length of 424L 1/2 or 8480 cells to be used in the end. It is not realistic to calculate on such a mesh.…”

Section: Unstable Two-dimensional Detonationmentioning

confidence: 99%

“…First, it was conducted three calculations on the rough mesh with h=0.2 till time t=40 when the flow structure is developed. In the first case we calculate throghout the domain of L dom ≈275L 1/2 (1375 cells) determined via (36). In the second case we set L dom =60L 1/2 (300 cells), and in the third case L dom =40L 1/2 (200 cells).…”

Section: Unstable Two-dimensional Detonationmentioning

confidence: 99%

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Second-order Godunov-type scheme for reactive flow calculations on moving meshes

Azarenok

Tang

2005

Journal of Computational Physics

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The method of calculating the system of gas dynamics equations coupled with the chemical reaction equation is considered. The flow parameters are updated in whole without splitting the system into a hydrodynamical part and an ODE part. The numerical algorithm is based on the Godunov's scheme on deforming meshes with some modification to increase the scheme-order in time and space. The variational approach is applied to generate the moving adaptive mesh. At every time step the functional of smoothness, written on the graph of the control function, is minimized. The grid-lines are condensed in the vicinity of the main solution singularities, e.g. precursor shock, fire zones, intensive transverse shocks, and slip lines, which allows resolving a fine structure of the reaction domain. The numerical examples relating to the Chapman-Jouguet detonation and unstable overdriven detonation are considered in both one and two space dimensions.

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“…This is done through the exact or approximate solution of the Riemann problem, giving rise to upwind methods [7,12,26,46,28]. For up-to-date background on these methods see, for example [11,39,20,17].…”

Section: Introductionmentioning

confidence: 99%

MUSTA fluxes for systems of conservation laws

Toro

Титарев

2006

Journal of Computational Physics

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Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Monographs and Surveys in Pure and Applied Mathematics, Vol. 118

Cited by 27 publications

References 0 publications

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

Second-order Godunov-type scheme for reactive flow calculations on moving meshes

MUSTA fluxes for systems of conservation laws

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