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Fourth‐order compact scheme for the one‐dimensional sine‐Gordon equation
Author(s) -
Cui Mingrong
Publication year - 2009
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20368
Subject(s) - mathematics , mathematical analysis , partial differential equation , boundary value problem , sine gordon equation , compact finite difference , dirichlet boundary condition , von neumann stability analysis , finite difference method , convergence (economics) , finite difference , ordinary differential equation , first order partial differential equation , neumann boundary condition , differential equation , nonlinear system , soliton , physics , quantum mechanics , economics , economic growth
Abstract Finite difference scheme to the generalized one‐dimensional sine‐Gordon equation is considered in this paper. After approximating the second order derivative in the space variable by the compact finite difference, we transform the sine‐Gordon equation into an initial‐value problem of a second‐order ordinary differential equation. Then Padé approximant is used to approximate the time derivatives. The resulting fully discrete nonlinear finite‐difference equation is solved by a predictor‐corrector scheme. Both Dirichlet and Neumann boundary conditions are considered in our proposed algorithm. Stability analysis and error estimate are given for homogeneous Dirichlet boundary value problems using energy method. Numerical results are given to verify the condition for stability and convergence and to examine the accuracy and efficiency of the proposed algorithm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009