A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain

Coclite, Giuseppe Maria; Ridder, J.K.; Risebro, Nils Henrik

doi:10.1007/s10543-016-0625-x

Cited by 23 publications

(41 citation statements)

References 30 publications

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“…We will now derive discrete versions of the entropy conditions (6) and (7). The entropy condition in the interior of the domain has already been proven in [7]. be defined by (14).…”

Section: Convergence Towards the Entropy Solutionmentioning

confidence: 99%

“…x −∞ u − ∞ −∞ u on the real line or P = x 0 u − 1 0 u in the unit interval; see [20,33,25,7]). Here we will consider the unit interval and choose P (0, t) = 0, which gives…”

Section: Introductionmentioning

confidence: 99%

“…[19, pp. 57-58], a function u ∈ C([0, T ]; L 1 (0, 1)) is an entropy solution if and only if inequalities (6) and (7) hold for all Kružkov entropy pairs,…”

Section: Introductionmentioning

confidence: 99%

“…This is the usual definition of entropy solutions of equation (1). However, regarding the entropy boundary condition instead of working with the original condition due to Bardos, le Roux and Nédélec [2], we will use the entropy boundary condition (7) introduced by Dubois and LeFloch [16]. Due to the regularizing effect of the P equation (4) we have that u ∈ L ∞ ((0, 1) × (0, T )) implies P [u] ∈ L ∞ (0, T ; W 1,∞ (0, 1)).…”

Section: Introductionmentioning

confidence: 99%

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A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions

Ridder

Ruf

2019

Bit Numer Math

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We prove convergence of a finite difference scheme to the unique entropy solution of a general form of the Ostrovsky-Hunter equation on a bounded domain with non-homogeneous Dirichlet boundary conditions. Our scheme is an extension of monotone schemes for conservation laws to the equation at hand. The convergence result at the center of this article also proves existence of entropy solutions for the initial-boundary value problem for the general Ostrovsky-Hunter equation. Additionally, we show uniqueness using Kružkov's doubling of variables technique. We also include numerical examples to confirm the convergence results and determine rates of convergence experimentally.

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“…We will now derive discrete versions of the entropy conditions (6) and (7). The entropy condition in the interior of the domain has already been proven in [7]. be defined by (14).…”

Section: Convergence Towards the Entropy Solutionmentioning

confidence: 99%

“…x −∞ u − ∞ −∞ u on the real line or P = x 0 u − 1 0 u in the unit interval; see [20,33,25,7]). Here we will consider the unit interval and choose P (0, t) = 0, which gives…”

Section: Introductionmentioning

confidence: 99%

“…[19, pp. 57-58], a function u ∈ C([0, T ]; L 1 (0, 1)) is an entropy solution if and only if inequalities (6) and (7) hold for all Kružkov entropy pairs,…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions

Ridder

Ruf

2019

Bit Numer Math

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“…An extra constraint for v is necessary to ensure the unique solution of the initial value problem (1.1). Referring to [7,22], there are two cases of constraints for v:…”

Section: The Dg Scheme For the Ov Equationmentioning

confidence: 99%

Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

Zhang

Xia

2020

J Sci Comput

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In this paper, we develop discontinuous Galerkin (DG) methods for the Ostrovsky-Vakhnenko (OV) equation, which yields the shock solutions and singular soliton solutions, such as peakon, cuspon and loop solitons. The OV equation has also been shown to have a bi-Hamiltonian structure. We directly develop the energy stable or Hamiltonian conservative discontinuous Galerkin (DG) schemes for the OV equation.Error estimates for the two energy stable schemes are also proved. For some singular solutions, including cuspon and loop soliton solutions, the hodograph transformation is adopted to transform the OV equation or the generalized OV system to the coupled dispersionless (CD) system. Subsequently, two DG schemes are constructed for the transformed CD system. Numerical experiments are provided to demonstrate the accuracy and capability of the DG schemes, including shock solution and, peakon, cuspon and loop soliton solutions.

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