## Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method.(English)Zbl 1096.65086

The author presents a finite difference scheme for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method [cf. D. Furihata, J. Comput. Phys. 156, 181–205 (1999; Zbl 0945.65103); Numer. Math 87, 675–699 (2001; Zbl 0974.65086)]. The stability, the existence and uniqueness and the convergence rate of the proposed finite difference scheme are shown.

### MSC:

 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 34G25 Evolution inclusions

### Citations:

Zbl 0945.65103; Zbl 0974.65086
Full Text:

### References:

 [1] D.G. Aronson, H.F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, vol. 446, pp. 5-49. · Zbl 0325.35050 [2] Cantrell, R.S.; Conser, C., Diffusive logistic equations with indefinite weights: population model in disrupted environments, Proc. roy. soc. Edinburgh section A, 112, 3-4, 293-318, (1989) · Zbl 0711.92020 [3] Chorin, A.J.; McCracken, M.F.; Hughes, T.J.R.; Marsden, J.E., Product formulas and numerical algorithms, Comm. pure appl. math., 31, 2, 205-256, (1978) · Zbl 0358.65082 [4] Du, Q.; Nicolaides, R.A., Numerical analysis of a continuum model of phase transition, SIAM J. numer. anal., 28, 5, 1310-1322, (1991) · Zbl 0744.65089 [5] Fei, Z.; Vázquez, L., Two energy conserving numerical schemes for the sine-Gordon equation, Appl. math. comput., 45, 17-31, (1991) · Zbl 0732.65107 [6] French, D.A.; Schaeffer, J.W., Continuous finite element methods which preserve energy properties for nonlinear problems, Appl. math. comput., 39, 3, 271-295, (1990) · Zbl 0716.65084 [7] D. Furihata, General derivation of finite difference schemes by means of a discrete variation, Doctoral Thesis, University of Tokyo, 1997 (in Japanese). [8] Furihata, D., Finite difference schemes $$\frac{\operatorname{\partial} u}{\operatorname{\partial} t} = \left(\frac{\operatorname{\partial}}{\operatorname{\partial} x}\right)^\alpha \frac{\delta G}{\delta u}$$ that inherit energy conservation or dissipation property, J. comput. phys., 156, 181-205, (1999) [9] Furihata, D., A stable and conservative finite difference scheme for the cahn – hilliard equation, Numer. math., 87, 675-699, (2001) · Zbl 0974.65086 [10] Greenspan, D., Conservative numerical methods for $$\ddot{x} = f(x)$$, J. comput. phys., 41, 28-41, (1984) · Zbl 0561.65056 [11] Hairer, E.; Lubich, C.; Wanner, G., Geometric numerical integration – structure – preserving algorithms for ordinary differential equations, (2002), Springer Berlin · Zbl 0994.65135 [12] Hirota, C.; Ide, T.; Okada, M.; Fukuoka, N., Generalized energy integrals and energy conserving numerical schemes for partial differential equations, Japan J. indust. appl. math., 21, 163-179, (2004) · Zbl 1064.65094 [13] Ide, T.; Hirota, C.; Okada, M., Generalized energy integral for $$\frac{\operatorname{\partial} u}{\operatorname{\partial} t} = \frac{\delta G}{\delta u}$$, its finite difference schemes by means of the discrete variational method and an application to Fujita problem, Adv. math. sci. appl., 12, 2, 755-778, (2002) · Zbl 1041.35027 [14] Iserles, A.; McLachlan, R.I.; Zanna, A., Approximately preserving symmetries in the numerical integration of ordinary differential equations, European J. appl. math., 10, 5, 419-445, (1999) · Zbl 0939.65139 [15] Ishimori, Y., Explicit energy conservative difference schemes for nonlinear dynamical systems with at most quadratic potentials, Phys. lett. A, 191, 5-6, 373-378, (1994) · Zbl 0960.70500 [16] Jimenez, S.; Pascual, P.; Aguirre, C.; Vázquez, L., A panoramic view of some perturbed nonlinear wave equation, Internat. J. bifurcation and chaos, 14, 1, 1-40, (2004) · Zbl 1063.65082 [17] Li, S.; Vu-Quoc, L., Finite difference calculus invariant structure of a class of algorithms for the nonlinear klein – gordon equation, SIAM J. numer. anal., 32, 1839-1875, (1995) · Zbl 0847.65062 [18] Maeda, S., Symplectic runge – kutta methods from the viewpoint of symmetry, J. math. tokushima univ., 34, 15-22, (2000) · Zbl 0985.65156 [19] T. Matsuo, Discrete variational method: its various extension and applications, Doctoral Thesis, University of Tokyo, 2002. [20] Matsuo, T.; Furihata, D., Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations, J. comput. phys., 171, 425-447, (2001) · Zbl 0993.65098 [21] Matsuo, T.; Furihata, D.; Sugihara, M.; Mori, M., Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method, Japan J. indust. appl. math., 19, 3, 311-330, (2002) · Zbl 1014.65083 [22] McLachlan, R.I.; Quispel, G.R.W.; Robidoux, N., Geometric integration using discrete gradients, Philos. trans. roy. soc. London A, 357, 1027-1045, (1999) · Zbl 0933.65143 [23] McLachlan, R.I., Spatial discretization of partial differential equations with integrals, IMA J. numer. anal., 23, 645-664, (2003) · Zbl 1080.65076 [24] McLachlan, R.I.; Quispel, G.R.W., Splitting methods, Acta numer., 11, 341-434, (2002) · Zbl 1105.65341 [25] Nakashima, K., Multi-layered stationary solutions for a spatially inhomogeneous allen – cahn equation, J. differential equations, 191, 1, 234-276, (2003) · Zbl 1034.34024 [26] Nakashima, K.; Tanaka, K., Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. inst. H. poincare anal. non lineaire, 20, 1, 107-143, (2003) · Zbl 1114.35005 [27] P.J. Olver, Application of Lie groups to Differential Equations, second ed., Graduate Texts in Mathematics, vol. 107, Springer, New York, 1993. · Zbl 0785.58003 [28] Olver, P.J., Geometric foundations of numerical algorithms and symmetry, Appl. algebra eng. comput. comm., 11, 417-436, (2001) · Zbl 0982.65135 [29] Rannacher, R., Finite element solution of diffusion problems with irregular data, Numer. math., 43, 309-327, (1984) · Zbl 0512.65082 [30] Sanz-Serna, J.M.; Calvo, M.P., Numerical Hamiltonian problems, (1994), Chapman & Hall London · Zbl 0816.65042 [31] Strauss, W.; Vázquez, L., Numerical solution of nonlinear klein – gordon equation, J. comput. phys., 28, 271-278, (1978) · Zbl 0387.65076 [32] Ueno, T.; Ide, T.; Okada, M., Wavelet collocation method for evolution equations with energy conserving property, Bull. sci. math., 127, 6, 569-583, (2003) · Zbl 1030.65104 [33] Wendlandt, J.M.; Marsden, J.E., Mechanical integrators derived from a discrete variational principle, Physica D, 106, 223-246, (1997) · Zbl 0963.70507 [34] Yoshida, H., Construction of higher order symplectic numerical scheme, Phys. lett. A, 87, 675-699, (1990)
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.