Flux form Semi-Lagrangian methods for parabolic problems

Luca Bonaventura; Roberto Ferretti

doi:10.1515/caim-2016-0022

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Flux form Semi-Lagrangian methods for parabolic problems

Communications in Applied and Industrial Mathematics

Volume 7 (2016): Issue 3 (September 2016)

By: Luca Bonaventura and Roberto Ferretti

Open Access

|Oct 2016

Abstract

A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and stability analysis is proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection diffusion and nonlinear parabolic problems.

References

1. M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, 2013.10.1137/1.9781611973051
Search in Google Scholar Back to article
2. M. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes., M2AN, vol. 29, pp. 97-122, 1995.10.1051/m2an/1995290100971
Search in Google Scholar Back to article
3. G. Milstein, The probability approach to numerical solution of nonlinear parabolic equations., Numerical Methods for Partial Differential Equations, vol. 18, pp. 490-522, 2002.10.1002/num.10020
Search in Google Scholar Back to article
4. G. Milstein and M. Tretyakov, Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations., Mathematics of Computation, vol. 69, pp. 237-567, 2000.10.1090/S0025-5718-99-01134-5
Search in Google Scholar Back to article
5. G. Milstein and M. Tretyakov, Numerical solution of the dirichlet problem for nonlinear parabolic equations by a probabilistic approach., IMA Journal of Numerical Analysis, vol. 21, pp. 887-917, 2001.10.1093/imanum/21.4.887
Search in Google Scholar Back to article
6. R. Ferretti, A technique for high-order treatment of diffusion terms in semi-Lagrangiaschemes., Communications in Computational Physics, vol. 8, pp. 445-470, 2010.10.4208/cicp.070709.011209a
Search in Google Scholar Back to article
7. J. Teixeira, Stable schemes for partial differential equations: The one-dimensional diffusion equation., Journal of Computational Physics, vol. 153, pp. 403-417, 1999.10.1006/jcph.1999.6283
Search in Google Scholar Back to article
8. L. Bonaventura, R. Redler, and R. Budich, Earth System Modelling 2: Algorithms, Code Infrastructure and Optimisation. New York: Springer Verlag, 2012.
Search in Google Scholar Back to article
9. L. Bonaventura and R. Ferretti, Semi-Lagrangian methods for parabolic problems in divergence form., SIAM Journal of Scientific Computing, vol. 36, pp. A2458 - A2477, 2014.10.1137/140969713
Search in Google Scholar Back to article
10. B. Leonard, A. Lock, and M. MacVean, Conservative explicit unrestricted-time-step multidimensional constancy-preserving advection schemes., Monthly Weather Review, vol. 124, pp. 2588-2606, November 1996.
Search in Google Scholar Back to article
11. S. Lin and R. B. Rood, Multidimensional flux-form semi-Lagrangian transport schemes., Monthly Weather Review, vol. 124, pp. 2046-2070, September 1996.
Search in Google Scholar Back to article
12. P. Frolkovic, Flux-based method of characteristics for contaminant transport in owing groundwater., Computing and Visualization in Science, vol. 5, pp. 73-83, 2002.10.1007/s00791-002-0089-1
Search in Google Scholar Back to article
13. E. S. N. Crouseilles, M. Mehrenberger, Conservative semi-Lagrangian schemes for the Vlasov equation., Journal of Computational Physics, vol. 229, pp. 1927-1953, 2010.
Search in Google Scholar Back to article
14. T. Phillips and A. Williams, Conservative semi-Lagrangian finite volume schemes., Numerical Methods for Partial Differential Equations, vol. 17, pp. 403-425, 2001.10.1002/num.1019
Search in Google Scholar Back to article
15. M. Restelli, L. Bonaventura, and R. Sacco, A semi-Lagrangian Discontinuous Galerkin method for scalar advection by incompressible flows, Journal of Computational Physics, vol. 216, pp. 195-215, 2006.10.1016/j.jcp.2005.11.030
Search in Google Scholar Back to article
16. J. Qiu and C. Shu, Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible ow., Journal of Computational Physics, vol. 230, p. 863ï ½889,201110.1016/j.jcp.2010.04.037
Search in Google Scholar Back to article
17. J. Qiu and C. Shu, Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov-Poisson system., Journal of Computational Physics, vol. 230, p. 8386ï ½8409, 2011.
Search in Google Scholar Back to article
18. J. Rossmanith and D. Seal, A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasovï ½ Poisson equations., Journal of Computational Physics, vol. 230, pp. 6203-6232, 2011.
Search in Google Scholar Back to article
19. J. Dukowicz, Conservative rezoning (remapping) for general quadrilateral meshes., Journal of Computational Physics, vol. 54, pp. 411-424, 1984.10.1016/0021-9991(84)90125-6
Search in Google Scholar Back to article
20. J. Laprise and R. Plante, A class of semi-Lagrangian integrated-mass (SLIM) numerical transport algorithms., Monthly Weather Review, vol. 123, pp. 553-565, 1995.10.1175/1520-0493(1995)123<;0553:ACOSLI>2.0.CO;2
Search in Google Scholar Back to article
21. L. Leslie and R. Purser, Three-dimensional mass-conserving semi-Lagrangian scheme employing foreward trajectories., Monthly Weather Review, vol. 123, pp. 2551-2566, 1995.
Search in Google Scholar Back to article
22. B. Machenhauer and M. Olk, The implementation of the semi-implicit scheme in cellintegrated semi-Lagrangian models., Atmosphere-Ocean, vol. XXXV, pp. 103-126, March 1997.10.1080/07055900.1997.9687344
Search in Google Scholar Back to article
23. M. Rancic, An efficient, conservative, monotone remapping for semi-Lagrangian transport algorithms., Monthly Weather Review, vol. 123, pp. 1213-1217, 1995.
Search in Google Scholar Back to article
24. J. Dukowicz and J. Baumgardner, Incremental remapping as a transport/advection algorithm., Journal of Computational Physics, vol. 160, pp. 318-335, 2000.10.1006/jcph.2000.6465
Search in Google Scholar Back to article
25. W. Lipscomb and T. Ringler, An incremental remapping transport scheme on a spherical geodesic grid., Monthly Weather Review, vol. 133, pp. 2335-2350, 2005.
Search in Google Scholar Back to article
26. R. Nair and B. Machenhauer, The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere., Monthly Weather Review, vol. 130, pp. 649-667, 2002.10.1175/1520-0493(2002)130<;0649:TMCCIS>2.0.CO;2
Search in Google Scholar Back to article
27. M. Zerroukat, N. Wood, and A. Staniforth, SLICE: a semi-Lagrangian inherently conserving and efficient scheme for transport problems., Quarterly Journal of the Royal Meteorological Society, vol. 128, pp. 2801-2820, 2002.
Search in Google Scholar Back to article
28. M. Zerroukat, N. Wood, and A. Staniforth, SLICE-S: A semi-Lagrangian inherently conserving and efficient scheme for transport problems on the sphere., Quarterly Journal of the Royal Meteorological Society, vol. 130, pp. 2649-2664, 2004.
Search in Google Scholar Back to article
29. M. Zerroukat, N. Wood, and A. Staniforth, A monotonic and positive-definite filter for a semi-Lagrangian inherently conserving and efficient (SLICE) scheme., Quarterly Journal of the Royal Meteorological Society, vol. 131, pp. 2923-2936, 2005.
Search in Google Scholar Back to article
30. M. Zerroukat, N. Wood, A. Staniforth, A. A. White, and J. Thuburn, An inherently conserving semi-implicit semi-Lagrangian discretisation of the shallow water equation on the sphere., Quarterly Journal of the Royal Meteorological Society, vol. 135, pp. 1104-1116, 2009.
Search in Google Scholar Back to article
31. R. Ferretti, Stability of some generalized Godunov schemes with linear high-order reconstructions., Journal of Scientific Computing, vol. 57, pp. 213-228, 2013.10.1007/s10915-013-9701-4
Search in Google Scholar Back to article
32. R. Ferretti, Equivalence of semi-Lagrangian and Lagrange-Galerkin schemes under constant advection speed., Journal of Computational Mathematics, vol. 28, pp. 461-473, 2010.10.4208/jcm.1003-m0012
Search in Google Scholar Back to article
33. E. Carlini, M. Falcone, and R. Ferretti, A Time - Adaptive Semi-Lagrangian Approximation to Mean Curvature Motion, in Numerical mathematics and advanced applications, pp. 732-739, Springer Berlin Heidelberg, 2006.10.1007/978-3-540-34288-5_71
Search in Google Scholar Back to article
34. G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh, vol. 16, no. 6, pp. 79-6, 1952.
Search in Google Scholar Back to article
35. G. Tumolo and L. Bonaventura, A semi-implicit, semi-Lagrangian, DG framework for adaptive numerical weather prediction., Quarterly Journal of the Royal Meteorological Society, vol. 141, pp. 2582-2601, 2015.
Search in Google Scholar Back to article
36. G. Tumolo, L. Bonaventura, and M. Restelli, A semi-implicit, semi-Lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations., Journal of Computational Physics, vol. 232, pp. 46-67, 2013.10.1016/j.jcp.2012.06.006
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.1515/caim-2016-0022 | Journal eISSN: 2038-0909

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Language: English

Page range: 56 - 73

Submitted on: May 29, 2015

Accepted on: Mar 20, 2016

Published on: Oct 1, 2016

Published by: Italian Society for Applied and Industrial Mathemathics

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Semi-Lagrangian methods,

Flux-Form Semi-Lagrangian methods,

Diffusion equations,

Divergence form

Related subjects:

Mathematics,

Numerical and computational mathematics,

Applied mathematics

© 2016 Luca Bonaventura, Roberto Ferretti, published by Italian Society for Applied and Industrial Mathemathics
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 7 (2016): Issue 3 (September 2016)