A Priori Error Estimate for a Well-balanced Scheme Designed for Inhomogeneous Scalar Conservation Laws

نویسنده

  • Laurent Gosse
چکیده

The aim of this note is to derive an a-priori error estimate for the approximate solution generated by means of the numerical scheme proposed in 5] and extensively studied in 4]. This scheme belongs to the class of well-balanced schemes introduced by Greenberg and LeRoux 6]; its main features are its robustness and its ability to preserve the correct steady-states of the continuous problem. Here, we show that it is endowed with the usual O(p x) error in most cases. Estimation d'erreur a priori pour un sch ema equilibre adapt e aux lois de conservation scalaires non-homog enes R esum e Le but de cette note est d'obtenir une estimation d'erreur a-priori pour les solutions approch ees g en er ees gr^ ace au sch ema num erique propos e dans 5] et etudi e dans 4]. Ce sch ema appartient a la classe des sch emas equilibre introduits par Greenberg et LeRoux 6]; ses principales caract eristiques sont sa robustesse et sa capacit e a pr eserver les etats stationnaires corrects du probl eme continu. Ici, nous montrons qu'il est ent^ ach e d'une erreur en O(p x) dans la plupart des cas. Version frann caise abr eg ee Dans cette note, on propose une m ethode permettant d'obtenir ais ement des estimations d'erreur a-priori pour les solutions approch ees des lois de conservtion scalaires avec termes sources (balance laws). Pour ce faire, on rappelle quelques el ements de la th eorie de Kruzkov 8] pour les solutions entropiques a variation born ee (Th eor eme 1). Le r esultat g en eral est enonc e dans le Th eor eme 2; celui-ci s'inspire des travaux ant erieurs de Bouchut et Perthame 2] puis de Kat-soulakis, Kossioris et Makridakis 7]. L'objectif est de quantiier l'erreur s eparant la solution exacte du probl eme consid er e d'avec une approximation v eriiant une certaine forme d'in egalit e d'entropie. Une application concerne la classe de sch emas num eriques \ equilibre" (well-balanced) introduits r ecemment par Greenberg et LeRoux 6] puis d evelopp es par exemple dans 4, 5]. En cons equence, on etablit dans le Th eor eme 3 que la vitesse de convergence de ce type d'approximations num eriques est le plus souvent identique a celle obtenue classiquement pour des probl emes homog enes.

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تاریخ انتشار 1998