### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 052, 23 pages      arXiv:1209.5028      https://doi.org/10.3842/SIGMA.2013.052
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

### Invariant Discretization Schemes Using Evolution-Projection Techniques

Alexander Bihlo a, b and Jean-Christophe Nave b
a) Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada
b) Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W., Montréal (QC) H3A 2K6, Canada

Received September 27, 2012, in final form July 28, 2013; Published online August 01, 2013

Abstract
Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy.

Key words: invariant numerical schemes; moving frame; evolution-projection method; heat equation.

pdf (639 kb)   tex (252 kb)

References

1. Bakirova M.I., Dorodnitsyn V.A., Kozlov R.V., Symmetry-preserving difference schemes for some heat transfer equations, J. Phys. A: Math. Gen. 30 (1997), 8139-8155, math.NA/0402367.
2. Bihlo A., Popovych R.O., Invariant discretization schemes for the shallow-water equations, SIAM J. Sci. Comput. 34 (2012), B810-B839, arXiv:1201.0498.
3. Bluman G.W., Kumei S., Symmetries and differential equations, Applied Mathematical Sciences, Vol. 81, Springer-Verlag, New York, 1989.
4. Bridges T.J., Reich S., Numerical methods for Hamiltonian PDEs, J. Phys. A: Math. Gen. 39 (2006), 5287-5320.
5. Budd C., Dorodnitsyn V., Symmetry-adapted moving mesh schemes for the nonlinear Schrödinger equation, J. Phys. A: Math. Gen. 34 (2001), 10387-10400.
6. Budd C.J., Huang W., Russell R.D., Moving mesh methods for problems with blow-up, SIAM J. Sci. Comput. 17 (1996), 305-327.
7. Budd C.J., Huang W., Russell R.D., Adaptivity with moving grids, Acta Numer. 18 (2009), 111-241.
8. Budd C.J., Iserles A., Geometric integration: numerical solution of differential equations on manifolds, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999), 945-956.
9. Cheh J., Olver P.J., Pohjanpelto J., Algorithms for differential invariants of symmetry groups of differential equations, Found. Comput. Math. 8 (2008), 501-532.
10. Chhay M., Hamdouni A., A new construction for invariant numerical schemes using moving frames, C. R. Mécanique 338 (2010), 97-101.
11. Chhay M., Hoarau E., Hamdouni A., Sagaut P., Comparison of some Lie-symmetry-based integrators, J. Comput. Phys. 230 (2011), 2174-2188.
12. Dawes A.S., Invariant numerical methods, Internat. J. Numer. Methods Fluids 56 (2008), 1185-1191.
13. Dorodnitsyn V., Applications of Lie groups to difference equations, Differential and Integral Equations and Their Applications, Vol. 8, CRC Press, Boca Raton, FL, 2011.
14. Dorodnitsyn V., Kozlov R., A heat transfer with a source: the complete set of invariant difference schemes, J. Nonlinear Math. Phys. 10 (2003), 16-50, math.AP/0309139.
15. Fels M., Olver P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
16. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
17. Fornberg B., A practical guide to pseudospectral methods, Cambridge Monographs on Applied and Computational Mathematics, Vol. 1, Cambridge University Press, Cambridge, 1996.
18. Frank J., Gottwald G., Reich S., A Hamiltonian particle-mesh method for the rotating shallow-water equations, in Meshfree Methods for Partial Differential Equations (Bonn, 2001), Lect. Notes Comput. Sci. Eng., Vol. 26, Editors M. Griebel, M.A. Schweitzer, T.J. Barth, M. Griebel, D.E. Keyes, R.M. Nieminen, D. Roose, T. Schlick, Springer, Berlin, 2003, 131-142.
19. Hairer E., Lubich C., Wanner G., Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics, Vol. 31, 2nd ed., Springer-Verlag, Berlin, 2006.
20. Huang W., Russell R.D., Adaptive moving mesh methods, Applied Mathematical Sciences, Vol. 174, Springer, New York, 2011.
21. Kim P., Invariantization of numerical schemes using moving frames, BIT 47 (2007), 525-546.
22. Kim P., Invariantization of the Crank-Nicolson method for Burgers' equation, Phys. D 237 (2008), 243-254.
23. Kim P., Olver P.J., Geometric integration via multi-space, Regul. Chaotic Dyn. 9 (2004), 213-226.
24. Leimkuhler B., Reich S., Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics, Vol. 14, Cambridge University Press, Cambridge, 2004.
25. Levi D., Winternitz P., Continuous symmetries of difference equations, J. Phys. A: Math. Gen. 39 (2006), R1-R63, nlin.SI/0502004.
26. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, 2nd ed., Springer-Verlag, New York, 1993.
27. Olver P.J., Geometric foundations of numerical algorithms and symmetry, Appl. Algebra Engrg. Comm. Comput. 11 (2001), 417-436.
28. Olver P.J., Generating differential invariants, J. Math. Anal. Appl. 333 (2007), 450-471.
29. Ovsiannikov L.V., Group analysis of differential equations, Academic Press Inc., New York, 1982.
30. Rebelo R., Valiquette F., Symmetry preserving numerical schemes for partial differential equations and their numerical tests, J. Difference Equ. Appl. 19 (2013), 738-757, arXiv:1110.5921.
31. Sommer M., Névir P., A conservative scheme for the shallow-water system on a staggered geodesic grid based on a Nambu representation, Q. J. R. Meteorol. Soc. 135 (2009), 485-494.
32. Staniforth A., Côté J., Semi-Lagrangian integration schemes for atmospheric models - a review, Mon. Weather Rev. 119 (1991), 2206-2223.
33. Stensrud D.J., Parameterization schemes: keys to understanding numerical weather prediction models, Cambridge University Press, Cambridge, 2007.
34. Stull R.B., An introduction to boundary layer meteorology, Atmospheric Sciences Library, Vol. 13, Kluwer Academic Publishers, Dortrecht, 1988.
35. Valiquette F., Winternitz P., Discretization of partial differential equations preserving their physical symmetries, J. Phys. A: Math. Gen. 38 (2005), 9765-9783, math-ph/0507061.