## THERMAL SCIENCE

International Scientific Journal

### COMPACT SCHEMES FOR KORTEWEG-DE VRIES EQUATION

**ABSTRACT**

This paper proposes one family of compact schemes for Korteweg-de Vries equation. In the deterministic case, the schemes are convergent with fourth-order accuracy both in space and in time. Moreover, the schemes are stable. The numerical dispersion relation is analyzed. We compare the schemes with one second-order scheme. The numerical examples test the effect of the schemes. In the stochastic case, we simulate the wave profile and three discrete dynamical quantities for Korteweg-de Vries equation with small noise. The white noise has stochastic influence on the profile and dynamical quantities of the solution. If the size of noise increases, the perturbation on the profile and dynamical quantities will increase accordingly.

**KEYWORDS**

PAPER SUBMITTED: 2016-06-13

PAPER REVISED: 2016-08-05

PAPER ACCEPTED: 2017-05-19

PUBLISHED ONLINE: 2017-09-09

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**4**, PAGES [1797 - 1806]

- Milstein, G., Tretyakov, M., Stochastic Numerics for Mathematical Physics, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1995
- Cui, Q. N., et al., Analytical and Numerical Methods for Thermal Science, Thermal Science, 20 (2016), 3, pp. IX-XIV
- Lu, J. F., Ma, L., Analytical Approach to a Generalized Hirota-Satsuma Coupled Korteweg-de Vries Equation by Modified Variational Iteration Method, Thermal Science, 20 (2016), 3, pp. 885-888
- He, J.-H., A New Fractal Derivation, Thermal Science, 15 (2011), Suppl. 1, pp. S145-S147
- Wang, Q., Homotopy Perturbation Method for Fractional KdV-Burgers Equation, Chaos Solitons & Fractals, 35 (2008), 5, pp. 843-850
- Zhang, S., et al., Multi-Wave Solutions for a Non-Isospectral KdV-Type Equation with Variable Coefficients, Thermal Science, 16 (2012), 15, pp. 1476-1479
- He, J.-H., Abdou, M. A., New Periodic Solutions for Nonlinear Evolution Equations Using Exp-Function Method, Chaos, Solitons & Fractals, 34 (2007), 5, pp. 1421-1429
- Soliman, A., Raslan, R., First Integral Method for the Improved Modified KdV Equation, International Journal of Nonlinear Science, 8 (2009), 1, pp. 11-18
- Lv, Z. Q., et al., A New Multi-Symplectic Scheme for the KdV Equation, Chinese Physics Letters, 28 (2011), 6, pp. 60205-60208
- Zhao, P. F., Qin, M. Z., Multisymplectic Geometry and Multisymplectic Preissmann Scheme for the KdV Equation, Journal of Physics A: Mathematical and General, 33 (2000), 18, pp. 3613-3626
- Canivar, A., et al., A Taylor-Galerkin Finite Element Method for the KdV Equation Using Cubic BSplines, Physica B Condensed Matter, 405 (2010), 16, pp. 3376-3383
- Shen, Q., A Meshless Method of Lines for the Numerical Solution of KdV Equation Using Radial Basis Functions, Engineering Analysis with Boundary Elements, 33 (2009), 10, pp. 1171-1180
- Hu, J., et al., Conservative Linear Difference Scheme for Rosenau-KdV Equation, Advances in Mathematical Physics, 2013 (2013), 2, pp. 64-64
- Lele, S., Compact Finite Difference Schemes with Spectral-Like Solution, Journal of Computational Physics, 103 (1992), 1, pp. 16-42
- Ma, Y. P., et al., High-Order Compact Splitting Multi-Symplectic Method for the Coupled Nonlinear Schroedinger Equations, Computer and Mathematics with Applications, 61 (2011), 2, pp. 319-333
- Sekhar, T., Raju, B., An Efficient Higher Order Compact Scheme to Capture Heat Transfer Solutions in Spherical Geometry, Computer Physics Communications, 183 (2012), 11, pp. 2337-2345
- Kanazawa, H., et al., A Conservative Compact Finite Difference Scheme for the KdV Equation, JSIAM Letters, 4 (2012), Mar., pp. 5-8
- Pan, X. T., et al., Numerical Analysis of a Pseudo-Compact C-N Conservative Scheme for the Rosenau-KdV Equation Coupling with the Rosenau-RLW Equation, Boundary Value Problems, 1 (2015), Dec., pp. 1-17
- Wang, Q., Variational Principle for Variable Coefficients KdV Equation, Physics Letters A, 358 (2006), 2, pp. 91-93
- Liu, X. S., et al., Dynamic Properties of the Cubic Nonlinear Schroedinger Equation by Symplectic Method, Chinese Physics, 15 (2005), 2, pp. 231-237
- Jiang, S. S., et al., Stochastic Multi-Symplectic Integrator for Stochastic Hamiltonian Nonlinear Schroedinger Equation, Communications in Computational Physics, 14 (2013), 2, pp. 393-411