On global solvability of initial value problem for hyperbolic Monge–Ampère equations and systems

Tunitsky, Dmitry Vasilievich

doi:10.1134/s1064562417040263

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…where δij is the Kronecker delta. Condition (13) guarantees the existence of solution to the system of linear equations on Aij . By applying the Tresse derivatives to differential invariants we get new differential invariants.…”

Section: Integrability Via Quotientsmentioning

confidence: 99%

“…The tendency of studying such phenomena is remaining nowadays as well, see, for example [7], where the case of Chaplygin gases is considered, [8,9], where the authors discuss weak shock waves, which is actually the case considered in the present paper, and also it is worth mentioning [10,11], where the influence of turbulence on detonations is investigated. The properties of global solvability for Euler equations and singularities of their solutions were also studied in a series of works [12,13,14,15].…”

Section: Introductionmentioning

confidence: 99%

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Singularities in one-dimensional Euler flows

Roop

2021

Preprint

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In this paper, a system of one-dimensional gas dynamics equations is considered. This system is a particular case of Jacobi type systems and has a natural representation in terms of 2-forms on 0-jet space. We use this observation to find a new class of multivalued solutions for an arbitrary thermodynamic state model and discuss singularities of their projections to the space of independent variables for the case of an ideal gas. Caustics and discontinuity lines are found.

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Section: Integrability Via Quotientsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Singularities in one-dimensional Euler flows

Roop

2021

Preprint

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“…On the other hand, the hyperbolic equation is more exotic and the literature is scarce. The papers [8,9] are the most notable references in the context of numerical results in illumination optics, and the papers [10,11] are the most important results regarding existence and uniqueness results. It is conjectured that designing optical systems using the hyperbolic equation allows for the construction of more compact optics.…”

Section: Introductionmentioning

confidence: 99%

Numerical Methods for the Hyperbolic Monge-Ampère Equation Based on the Method of Characteristics

Bertens¹,

Vugts²,

Anthonissen³

et al. 2021

Preprint

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We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation. The MOC gives rise to two mutually coupled systems of ordinary differential equations. As a special case we consider the Monge-Ampère (MA) equation, for which we solve the system of ODE's using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Numerical examples demonstrate the performance of the methods.

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Singularities in one-dimensional Euler flows

Roop

2021

Journal of Geometry and Physics

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On global solvability of initial value problem for hyperbolic Monge–Ampère equations and systems

Cited by 4 publications

References 3 publications

Singularities in one-dimensional Euler flows

Singularities in one-dimensional Euler flows

Numerical Methods for the Hyperbolic Monge-Ampère Equation Based on the Method of Characteristics

Singularities in one-dimensional Euler flows

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