Advertisement

An Extrapolation Method of Crank-Nicolson Finite Difference Scheme for Distributed Control Equation

Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 217)

Abstract

This paper studies a finite difference scheme for the distributed control equation, which is based on Crank-Nicolson finite difference scheme and is constructed by applying an extrapolation technique to the nonlinear term. We proved the existence, uniqueness and convergence of the numerical solution. In literature review, there is no report of theoretical studies about the extrapolation. Meanwhile, these theoretical studies are confirmed by numerical experiments in the end. These show that the scheme is a practical numerical method for some computations or numerical simulations which require less accuracy and less computational time.

Keywords

Extrapolation method Finite difference scheme Distributed control equation Existence Uniqueness Convergence 

References

  1. 1.
    Hsu SB, Hwang TW, Kuang Y (2003) A ratio dependent food chain model and its applications to biological control. Math Biosci 181:55–83MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Wang M (1993) Nonlinear parabolic equations, vol 57. Science Press, Beijing, pp 35–37Google Scholar
  3. 3.
    Yuan GW, Shen LJ, Zhou Y (2001) Unconditional stability of parallel alternating difference schemes for semi-linear parabolic systems. Appl Math Comput 117:267–283MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Voss DA, Khaliq AQM (1999) A linearly implicit predictor-correct method for reaction-diffusion equations. Comput Math Appl 38:207–216MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Wu H (2006) A difference scheme for semi-linear parabolic equations regarding the distributed control. Math Appl 19(4):827–834MathSciNetMATHGoogle Scholar
  6. 6.
    Sun Z (1994) A class of second-order accurate difference schemes for quasi-linear parabolic equations. J Comp Math 16(4):347–361MATHGoogle Scholar
  7. 7.
    Chang Q, Jia E, Sun W (1999) Difference schemes for solving the generalized nonlinear Schrodinger equation. J Comput Phys 148:397–415MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Zhou J, Hu B (2011) A kind of extrapolation finite difference scheme with non-uniform mesh for semi-linear parabolic equation. J Sichuan Univ (Ed Nat Sci) 48(5):1018–1022Google Scholar
  9. 9.
    Sun Z (2004) Numerical methods for partial differential equations. Science Press, BeijingGoogle Scholar
  10. 10.
    Zhou Y (1990) Application of discrete functional analysis to the finite difference method, vol 734. Inter Acad Publishers, Beijing, pp 25–26Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceYangtze Normal UniversityChongqingChina
  2. 2.Department of Basic TeachingChengdu Aeronautic Vocational and Technical CollegeChengduChina

Personalised recommendations