×

RKMK method of solving non-damping LL equations and ferromagnet chain equations. (English) Zbl 1058.65099

Summary: We introduce a kind of Lie group method to solve non-damping Landau-Lifshitz (LL) equations and the ferromagnet chain equations. The basic idea is to discretize the partial differential equation into the form \(dY/dt=F(t,Y)\), then apply the Lie group method to it. The iterative numerical algorithm can preserve the property of square conservation of the discrete equation. The algorithm has the same accuracy as the classical Runge-Kutta method. It shows that we have proposed a valid algorithm to solve ferromagnet chain equations. The method can be applied to other similar partial differential equations.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Munthe-Kaas, H., High order runge – kutta methods on manifolds, Appl. numer. math, 29, 115-127, (1999) · Zbl 0934.65077
[2] Calvo, M.P.; Iserles, A.; Zanna, A., Runge – kutta methods for orthogonal and isospectral flows, Appl. numer. math, 22, 153-163, (1996) · Zbl 0871.65074
[3] Iserles, A.; Munthe-Kaas, H.; Nørsett, S.P., Lie group methods, Acta numer, 215-365, (2000) · Zbl 1064.65147
[4] Qin, M.Z., Two difference schemes for the systems of ferromagnetic chain, J. comp. math, 11, (1986)
[5] Crouch, P.E.; Grossman, R., Numerical integration of ordinary differential equations on manifolds, J. nonlinear sci, 3, 1-33, (1993) · Zbl 0798.34012
[6] Zanna, A., Collocation and relaxed collocation for the fer and the Magnus expansion, SIAM J. numer. anal, 36, 1145-1182, (1999) · Zbl 0936.65092
[7] Budd, C.J.; Iserles, A., Geometric integration: numerical solution of differential equations on manifolds, Philos. trans. R. soc. lond., A, 357, 945-956, (1999) · Zbl 0933.65142
[8] J.A.G. Roberts, C.J. Thompson, Dynamics of the classical Heisenberg spin chain, Technique Report 11-1978, Mathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia · Zbl 0662.35102
[9] S. Krogstad, A low complexity Lie group method on the Stiefel manifold, report no, July 2001, department of informatics, University of Bergen, Bergen, Norway
[10] Iserles, A., Numerical methods on (and off) manifolds, (), 180-189 · Zbl 0869.65047
[11] Varadarajan, V.S., Lie groups, Lie algebras and their representations, (1984), Springer-Verlag Berlin · Zbl 0955.22500
[12] A. Marthinsen, H. Munthe-Kaas, B. Owren, Simulation of ordinary differential equations on manifolds, Department of Mathemathematical Science, Num, N-7304, Trondheim, Norway · Zbl 0870.34015
[13] Munthe-Kaas, H., Runge – kutta methods on Lie groups, Bit, 38, 92-111, (1996) · Zbl 0904.65077
[14] Chen, J.B.; Munthe-Kass, H.; Qin, M.Z., Square-conservation scheme for a class of evolution equations using Lie-group methods, SIAM J. numer. anal, 39, 6, 216-2178, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.