The Green’s function of the Lax–Wendroff and Beam–Warming schemes

Coulombel, Jean-François

Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 2, pp. 247-294.

On obtient une borne Gaussienne généralisée pour la fonction de Green des schémas de Lax–Wendroff et Beam–Warming. Cette borne permet de préciser la région de l’espace qui conduit à l’instabilité bien connue de ces schémas pour la norme uniforme. On retrouve par ailleurs des bornes uniformes quand ces schémas sont appliqués à des suites à variations bornées.

We prove a sharp uniform generalized Gaussian bound for the Green’s function of the Lax–Wendroff and Beam–Warming schemes. Our bound highlights the spatial region that leads to the well-known (rather weak) instability of these schemes in the maximum norm. We also recover uniform bounds in the maximum norm when these schemes are applied to initial data of bounded variation.

Classification : 65M06, 65M12, 35L02
Keywords: Transport equation, Lax–Wendroff scheme, Beam–Warming scheme, difference approximation, convolution, stability, local limit theorem
Mot clés : Équation de transport, schéma de Lax–Wendroff, schéma de Beam–Warming, approximation par différences finies, convolution, stabilité, théorème de la limite locale

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