Construction and convergence study of schemes preserving the elliptic local maximum principle

Jerome Droniou, Christophe Le Potier

Research output: Contribution to journalArticleResearchpeer-review

49 Citations (Scopus)

Abstract

We present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models.
Original languageEnglish
Pages (from-to)459 - 490
Number of pages32
JournalSIAM Journal on Numerical Analysis
Volume49
Issue number2
DOIs
Publication statusPublished - 2011
Externally publishedYes

Cite this

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Construction and convergence study of schemes preserving the elliptic local maximum principle. / Droniou, Jerome; Le Potier, Christophe.

In: SIAM Journal on Numerical Analysis, Vol. 49, No. 2, 2011, p. 459 - 490.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Droniou, Jerome

AU - Le Potier, Christophe

PY - 2011

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AB - We present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models.

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