On the convergence of WENO schemes for scalar hyperbolic conservation laws

Abstract 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 $$TV(u^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is uniformly bounded for all n , k with $$k&lt;k_{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> , $$nk\le T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> . 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).