WENO schemes are high order accurate shock capturing methods for hyperbolic conservation laws. In this paper theoretical stability properties are presented. In fact, we prove, assuming some reasonable hypotheses from a practical point of view, that the numerical flux is Lipschitz continuous and that the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_909_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$TV(u^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>V</mi> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is uniformly bounded for all n, k with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_909_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$k<k_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo><</mo> <msub> <mi>k</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_909_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$nk\le T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mi>k</mi> <mo>≤</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>. Both results give the two sufficient properties used in the convergence framework presented in Leveque (Numerical methods for conservation laws. Birkhäuser Verlag (Lectures in Mathematics), 1990).

On the convergence of WENO schemes for scalar hyperbolic conservation laws

期刊论文
出版模式：

开放获取
发表日期：

2025年4月16日

查看全文

Sergio Amat,
Sonia Busquier,
Juan Ruiz

摘要

WENO schemes are high order accurate shock capturing methods for hyperbolic conservation laws. In this paper theoretical stability properties are presented. In fact, we prove, assuming some reasonable hypotheses from a practical point of view, that the numerical flux is Lipschitz continuous and that the $TV(u^n)$ $T V (u^{n})$ is uniformly bounded for all n, k with $k<k_{0}$ $k < k_{0}$ , $nk\le T$ $n k \leq T$ . Both results give the two sufficient properties used in the convergence framework presented in Leveque (Numerical methods for conservation laws. Birkhäuser Verlag (Lectures in Mathematics), 1990).