Non linear schemes for the heat equation in 1D∗
1 LJLL, UPMC,
2 Laboratoire Jacques-Louis Lions University Pierre et Marie Curie Boîte courrier 187 75252 Paris Cedex 05 France.
Revised: 9 June 2013
Inspired by the growing use of non linear discretization techniques for the linear diffusion equation in industrial codes, we construct and analyze various explicit non linear finite volume schemes for the heat equation in dimension one. These schemes are inspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 691–695]. They preserve the maximum principle and admit a finite volume formulation. We provide a original functional setting for the analysis of convergence of such methods. In particular we show that the fourth discrete derivative is bounded in quadratic norm. Finally we construct, analyze and test a new explicit non linear maximum preserving scheme with third order convergence: it is optimal on numerical tests.
Mathematics Subject Classification: 65J05 / 65M08 / 65M12
Key words: Finite volume schemes / heat equation / non linear correction
The author acknowledges the support of ANR under contract ANR-12-BS01-0006-01. Moreover this work was carried out within the framework of the European Fusion Development Agreement and the French Research Federation for Fusion Studies. It is supported by the European Communities under the contract of Association between Euratom and CEA. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
© EDP Sciences, SMAI 2013