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BIT Numerical Mathematics
Published in: BIT Numerical Mathematics 3/2019

13-03-2019

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

Authors: J. Ridder, A. M. Ruf

Published in: BIT Numerical Mathematics | Issue 3/2019

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Abstract

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.
previous article On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case
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Footnotes
1
Here, we have https://static-content.springer.com/image/art%3A10.1007%2Fs10543-019-00746-7/MediaObjects/10543_2019_746_IEq234_HTML.gif and therefore \(\lambda =\Delta t/\Delta x\) should satisfy \(\lambda \le 36\). However, since the https://static-content.springer.com/image/art%3A10.1007%2Fs10543-019-00746-7/MediaObjects/10543_2019_746_IEq237_HTML.gif bound from Lemma 1 allows for some growth of https://static-content.springer.com/image/art%3A10.1007%2Fs10543-019-00746-7/MediaObjects/10543_2019_746_IEq238_HTML.gif choosing a smaller \(\lambda \) can be neccessary.
 
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Metadata
Title
A convergent finite difference scheme for the Ostrovsky–Hunter equation with Dirichlet boundary conditions
Authors
J. Ridder
A. M. Ruf
Publication date
13-03-2019
Publisher
Springer Netherlands
Published in
BIT Numerical Mathematics / Issue 3/2019
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-019-00746-7

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