Numerical solution of damped nonlinear Klein–Gordon equations using variational method and finite element approach

• Published in: Applied mathematics and computation Vol. 162; no. 1; pp. 381 - 401
• Main Authors: Wang, QuanFang, Cheng, DaiZhan
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 04-03-2005; Elsevier
• Subjects: Approximations and expansions; Calculus of variations and optimal control; Exact sciences and technology; Finite element methods; Gauss–Legendre quadrature; Klein–Gordon equations; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Numerical solution; Real functions; Runge–Kutta method; Sciences and techniques of general use; Gauss–Legendre quadrature; Runge–Kutta method; Finite element methods; Numerical solution; Klein–Gordon equations; Three-dimensional calculations; Numerical integration; Differential equation; Continuous time; Continuous function; Runge Kutta method; Algorithm; Numerical approximation; Non linear equation; Finite element method; Graphics; Extrapolation; Interpolation; Numerical analysis; Klein-Gordon equations; Variational equation; Multistep method; Cubature; Applied mathematics; Quadrature formula; Gauss-Legendre quadrature
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2003.12.102

Numerical treatment for damped nonlinear Klein–Gordon equations, based on variational method and finite element approach, is studied. A semi-discrete algorithm is proposed by using quadratic interpolation functions of continuous time and spatial dimension one. The Gauss–Legendre quadrature has been utilized for numerical integrations of nonlinear terms, and Runge–Kutta method is used for solving ordinary differential equation. Finally, three dimensional graphics of numerical solutions are used to demonstrate the numerical results.