Discrete variational derivative method—A structure-preserving numerical method for partial differential equations

Furihata, Daisuke; Matsuo, Takayasu

doi:10.1090/suga/435

Discrete variational derivative method—A structure-preserving numerical method for partial differential equations
HTML articles powered by AMS MathViewer

by Daisuke Furihata and Takayasu Matsuo
Translated by: Daisuke Furihata and Takayasu Matsuo

Sugaku Expositions 31 (2018), 231-255

DOI: https://doi.org/10.1090/suga/435

Published electronically: September 19, 2018

PDF | Request permission

Abstract:

In these decades structure-preserving numerical methods have been developed widely. In general, “structure-preserving” means that the numerical and discrete scheme inherits some mathematical properties from the original differential equation. It means that the design process of a structure-preserving method is a framework to discretize some mathematical properties of equations.

This manuscript describes discrete variational derivative method, which is one of structure-preserving methods for partial/ordinary differential equations. The discrete variational derivative method is a method to design some numerical schemes that inherit some dissipative or conservative properties via discretization of some relationships between the properties and variational structures of ordinary differential equations. We explain some basic concepts of a discrete variational derivative method with some typical examples and show some recent works based on it.

References

Allen, S. M. and Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085–1095.
W. F. Ames, Nonlinear partial differential equations in engineering, Academic Press, New York-London, 1965. MR 0210342
Baillon, J.B. and Chadam, J.M., The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, edited by G.M. de la Penha and L.A. Medeiros, North Holland, Amsterdam, 1978.
Bo Ling Guo, The global solution of the system of equations for a complex Schrödinger field coupled with a Boussinesq-type self-consistent field, Acta Math. Sinica 26 (1983), no. 3, 295–306 (Chinese). MR 721681
J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, Providence, RI, 1999. MR 1691575, DOI 10.1090/coll/046
Thomas J. Bridges and Sebastian Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A 284 (2001), no. 4-5, 184–193. MR 1854689, DOI 10.1016/S0375-9601(01)00294-8
C. J. Budd and M. D. Piggott, Geometric integration and its applications, Handbook of numerical analysis, Vol. XI, Handb. Numer. Anal., XI, North-Holland, Amsterdam, 2003, pp. 35–139. MR 2009771
Cahn, J.W. and Hilliard, J.E., Free energy of a non-uniform system. I. interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.
G. F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945), 157–165. MR 12351, DOI 10.1090/S0033-569X-1945-12351-4
David Cohen, Brynjulf Owren, and Xavier Raynaud, Multi-symplectic integration of the Camassa-Holm equation, J. Comput. Phys. 227 (2008), no. 11, 5492–5512. MR 2414917, DOI 10.1016/j.jcp.2008.01.051
Qiang Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal. 53 (1994), no. 1-2, 1–17. MR 1379180, DOI 10.1080/00036819408840240
Yukiyoshi Ebihara, Spherically symmetric solutions of some semilinear wave equations, Nonlinear mathematical problems in industry, II (Iwaki, 1992) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 2, Gakk\B{o}tosho, Tokyo, 1993, pp. 431–434. MR 1370482
Eguchi, T., Oki, K., and Matsumura, S., Kinetics of ordering with phase separation, Mat. Res. Soc. Symp. Proc., 21 (1984), 589–594.
Erwan Faou, Ernst Hairer, and Truong-Linh Pham, Energy conservation with non-symplectic methods: examples and counter-examples, BIT 44 (2004), no. 4, 699–709. MR 2211040, DOI 10.1007/s10543-004-5240-6
Fermi, E., Pasta, J. and Ulam, S., Studies of nonlinear problems I, Los Alamos Sci. Lab. Rept., LA-1490, 1955.
Fock, V.A., Zur Schrödingerschen Wellenmechanik, Z. Phys. A, 38 (1926), 242–250.
Fock, V.A., Über die Invariante Form der Wellen- und der Bewegungsgleichungen für einen Geladenen Massenpunkt, Z. Phys. A, 39 (1926), 226–232.
Furihata, D. and Mori, M., A stable finite difference scheme for the Cahn–Hilliard equation based on the Lyapunov functional, ZAMM Z. angew. Math. Mech., 76 (1996), 405–406.
Daisuke Furihata, Finite difference schemes for $\partial u/\partial t=(\partial /\partial x)^\alpha \delta G/\delta u$ that inherit energy conservation or dissipation property, J. Comput. Phys. 156 (1999), no. 1, 181–205. MR 1727636, DOI 10.1006/jcph.1999.6377
Daisuke Furihata and Takayasu Matsuo, Discrete variational derivative method, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011. A structure-preserving numerical method for partial differential equations. MR 2744841
Gibbons, J., Thornhill, S.G., Wardrop, M.J., and Ter Haar, D., On the theory of Langmuir solitons, J. Plasma Phys., 17 (1977), 153–170.
O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci. 6 (1996), no. 5, 449–467. MR 1411343, DOI 10.1007/s003329900018
O. Gonzalez, Exact energy and momentum conserving algorithms for general models in nonlinear elasticity, Comput. Methods Appl. Mech. Engrg. 190 (2000), no. 13-14, 1763–1783. MR 1807477, DOI 10.1016/S0045-7825(00)00189-4
Gordon, W., Der Comptoneffekt nach der Schrödingerschen Theorie, Z. Phys. A, 40 (1926), 117–133.
Donald Greenspan, Discrete models, Applied Mathematics and Computation, No. 3, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1973. MR 0366132
E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento (10) 20 (1961), 454–477 (English, with Italian summary). MR 128907
Ernst Hairer, Long-time energy conservation of numerical integrators, Foundations of computational mathematics, Santander 2005, London Math. Soc. Lecture Note Ser., vol. 331, Cambridge Univ. Press, Cambridge, 2006, pp. 162–180. MR 2277105, DOI 10.1017/CBO9780511721571.005
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations. MR 1904823, DOI 10.1007/978-3-662-05018-7
Jialin Hong, Shanshan Jiang, and Chun Li, Explicit multi-symplectic methods for Klein-Gordon-Schrödinger equations, J. Comput. Phys. 228 (2009), no. 9, 3517–3532. MR 2517531, DOI 10.1016/j.jcp.2009.02.006
Jialin Hong, Hongyu Liu, and Geng Sun, The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs, Math. Comp. 75 (2006), no. 253, 167–181. MR 2176395, DOI 10.1090/S0025-5718-05-01793-X
Takanori Ide, Chiaki Hirota, and Masami Okada, Generalized energy integral for ${\partial u\over \partial t}={\delta G\over \delta u}$, its finite difference schemes by means of the discrete variational method and an application to Fujita problem, Adv. Math. Sci. Appl. 12 (2002), no. 2, 755–778. MR 1943989
Salvador Jiménez, Derivation of the discrete conservation laws for a family of finite difference schemes, Appl. Math. Comput. 64 (1994), no. 1, 13–45. MR 1285464, DOI 10.1016/0096-3003(94)90137-6
Fritz John, Partial differential equations, 3rd ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York-Berlin, 1978. MR 514404
Keller, F. F. and Segel, L. A., Initiation of slime mold aggregation viewed as an instability, J. Theor. Bio., 26 (1970), 399–415.
Klein, O., Quantentheorie und Fünfdimensionale Relativitätstheorie, Z. Phys. A, 37 (1926), 895–906.
D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. (5) 39 (1895), no. 240, 422–443. MR 3363408, DOI 10.1080/14786449508620739
Benedict Leimkuhler and Sebastian Reich, Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004. MR 2132573
C. David Levermore and Marcel Oliver, The complex Ginzburg-Landau equation as a model problem, Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994) Lectures in Appl. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1996, pp. 141–190. MR 1363028, DOI 10.1080/03605309708821254
Christian Lubich, From quantum to classical molecular dynamics: reduced models and numerical analysis, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2474331, DOI 10.4171/067
V. S. Manoranjan, T. Ortega, and J. M. Sanz-Serna, Soliton and antisoliton interactions in the “good” Boussinesq equation, J. Math. Phys. 29 (1988), no. 9, 1964–1968. MR 957220, DOI 10.1063/1.527850
Jerrold E. Marsden, George W. Patrick, and Steve Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys. 199 (1998), no. 2, 351–395. MR 1666871, DOI 10.1007/s002200050505
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer. 10 (2001), 357–514. MR 2009697, DOI 10.1017/S096249290100006X
Takayasu Matsuo, Masaaki Sugihara, Daisuke Furihata, and Masatake Mori, Linearly implicit finite difference schemes derived by the discrete variational method, Sūrikaisekikenkyūsho K\B{o}kyūroku 1145 (2000), 121–129. Numerical solution of partial differential equations and related topics (Japanese) (Kyoto, 1999). MR 1795452
Matsuo, T., Sugihara, M., and Mori, M., A derivation of a finite difference scheme for the nonlinear Schrödinger equation, advances in numerical mathematics; Proceedings of the fourth Japan-China joint seminar on numerical mathematics (Chiba, 1998), 243–250, GAKUTO Internat. Ser. Math. Sci. Appl., 12, Gakkotosho, Tokyo, 1999.
Robert I. McLachlan, G. R. W. Quispel, and Nicolas Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999), no. 1754, 1021–1045. MR 1694701, DOI 10.1098/rsta.1999.0363
Robert I. McLachlan and Nicolas Robidoux, Antisymmetry, pseudospectral methods, and conservative PDEs, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 994–999. MR 1870274
Alan C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (1969), no. 2, 279–303. MR 3363403, DOI 10.1017/S0022112069000176
Pitaevskii, L. P., Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, 13 (1961), 451–454.
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation, vol. 7, Chapman & Hall, London, 1994. MR 1270017
Shimoji, S. and Kawai, T., Multi-valued solution to nonlinear wave equations (in Japanese), in the proceedings of Numerical Analysis Symposium (Nagano, 1997), 39–42.
Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311
Masuo Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A 146 (1990), no. 6, 319–323. MR 1059400, DOI 10.1016/0375-9601(90)90962-N
Swift, J. and Hohenberg, P. C., Hydrodynamic fluctuations at the convective instability, Physical Review A, 15 (1977), 319–328.
Tanaka, H., and Nishi, T., Direct determination of the probability distribution function of concentration in polymer mixtures undergoing phase separation, Phys. Rev. Lett., 59 (1987), 692–695.
Takaharu Yaguchi, Takayasu Matsuo, and Masaaki Sugihara, An extension of the discrete variational method to nonuniform grids, J. Comput. Phys. 229 (2010), no. 11, 4382–4423. MR 2609783, DOI 10.1016/j.jcp.2010.02.018
Takaharu Yaguchi, Takayasu Matsuo, and Masaaki Sugihara, Conservative numerical schemes for the Ostrovsky equation, J. Comput. Appl. Math. 234 (2010), no. 4, 1036–1048. MR 2609558, DOI 10.1016/j.cam.2009.03.008
Haruo Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A 150 (1990), no. 5-7, 262–268. MR 1078768, DOI 10.1016/0375-9601(90)90092-3
Yoshinaga, T., Chaos in water waves due to the long-short wave interaction (in Japanese), Butsuri, 47 (1992), 300–303.
Zakharov, V.E., Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908–914.
Zakharov, V. E. and Kuznetsov, E. A., Three-dimensional solitons, Sov. Phys. JETP, 39 (1974), 285–286.