Non linear schemes for the heat equation in 1D
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 1, pp. 107-134.

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.

DOI : 10.1051/m2an/2013096
Classification : 65J05, 65M08, 65M12
Mots-clés : finite volume schemes, heat equation, non linear correction 
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     title = {Non linear schemes for the heat equation in {1D}},
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Després, Bruno. Non linear schemes for the heat equation in 1D. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 1, pp. 107-134. doi : 10.1051/m2an/2013096. http://www.numdam.org/articles/10.1051/m2an/2013096/

[1] I. Aavatsmark, T. Barkve, O. Boe, T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700-1716. | MR | Zbl

[2] F. Boyer, F. Hubert, Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032-3070. | MR | Zbl

[3] F. Brezzi, K. Lipnikov, M. Shashkov, V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Meth. Appl. Mech. Eng. 196 (2007) 3682-3692. | MR | Zbl

[4] C. Buet, B. Després and E. Franck, Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes Numerische Mathematik, Online First (2012). | MR | Zbl

[5] C. Cancès, M. Cathala, C. Le Potier, Monotone coercive cell-centered finite volume schemes for anisotropic diffusion equations, online Numer. Math. (2013). | MR | Zbl

[6] G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations. Springer-Verlag (2001) | MR | Zbl

[7] B. Després, Convergence of non-linear finite volume schemes for linear transport. In Notes from the XIth Jacques-Louis Lions Hispano-French School on Numerical Simulation in Physics and Engineering. Grupo Anal. Teor. Numer. Modelos Cienc. Exp. Univ. Cadiz (2004) 219-239. | MR | Zbl

[8] B. Després, Lax theorem and Finite Volume schemes. Math. Comput. 73 (2004) 1203-1234. | MR | Zbl

[9] J. Droniou, C. Le Potier, Construction and convergence study of local-maximum-principle preserving schemes for elliptic equations. SIAM J. Numer. Anal. 49 (2011) 459-490. | MR | Zbl

[10] L.C. Evans, Partial Differential Equations. Rhode Island: American Mathematical Society, Providence (1988). | MR | Zbl

[11] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, vol. 7 of Handbook of Numerical Analysis. Edited by P.G. Ciarlet and J.L. Lions. North Holland (2000) 713-1020. | Zbl

[12] R. Eymard, T. Gallouët and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces, IMA J Numer Anal. 30 (2010) 1009-1043. | MR | Zbl

[13] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, New York (1983). | MR | Zbl

[14] A. Genty and C. Le Potier, Maximum and minimum principles for radionuclide transport calculations in geological radioactive waste repository: comparison between a mixed hybrid finite element method and finite volume element discretizations. Transp. Porous Media 88 (2011) 65-85. | MR

[15] E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, vol. 118 of Applied Mathematical Sciences. Springer (1996). | MR | Zbl

[16] R. Herbin, F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in: 5th International Symposium on Finite Volumes for Complex Applications, edited by V.R. Eymard and J.M. Herard. Wiley (2008) 659-692. | MR | Zbl

[17] Hermeline F., A finite volume method for approximating 3D diffusion operators on general meshes. J. Comput. Phys. 228 (2009) 5763-5786. | MR | Zbl

[18] Kershaw D., Differencing of the diffusion equation in Lagrangian hydrodynamic codes. J. Comput. Phys. 39 (1981) 375-395. | MR | Zbl

[19] C. Le Potier, Correction non linéaire et principe du maximum pour la discrétisation d'opérateurs de diffusion avec des schémas volumes finis centrés sur les mailles. C. R. Acad. Sci. Paris, Ser. I 348 (2010) 691-695. | MR | Zbl

[20] C. Lepotier, private communication (2012).

[21] R.J. Leveque, Numerical Methods for Conservation Laws, Lectures in Mathematics. ETH-Zurich Birkhauser-Verlag, Basel (1990). | MR | Zbl

[22] K. Lipnikov, M. Shashkov, I. Yotov, Local flux mimetic finite difference methods. Numer. Math. 112 (2009) 115-152. | MR | Zbl

[23] K. Lipnikov and M. Shashkov, A framework for developing a mimetic tensor artificial viscosity for Lagrangian hydrocodes on arbitrary polygonal meshes. J. Comput. Phys. 229 (2010) 7911-7941. | MR

[24] K. Lipnikov, G. Manzini and D. Svyatskiy, Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230 (2011) 2620-2642. | MR | Zbl

[25] P.L. Roe, Characteristic-based schemes for the Euler equations. Ann. Rev. Fluid Mech. 18 (1986) 337-365. | MR | Zbl

[26] Z. Sheng, J. Yue, G. Yuan, Monotone Finite volume schemes of non-equilibrium radiation diffusion equations of distorted meshes, SIAM J. Sci. Comput. 31 (2009) 2915-2934. | MR | Zbl

[27] Yu.I. Shokin, The method of differential approximation, Springer-Verlag (1983). | MR | Zbl

[28] P. Sweby, High-resolution schemes using flux limiters for hyperbolic conservation-laws. SIAM J. Numer. Anal. 21 (1984) 995-1011. | MR | Zbl

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