A class of one-step time integration schemes for second-order hyperbolic differential equations

• Published in: Mathematical and computer modelling Vol. 33; no. 4; pp. 431 - 443
• Main Authors: Chawla, M.M., Al-Zanaidi, M.A.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.02.2001; Elsevier Science
• Subjects: Exact sciences and technology; Extended one-step time integration schemes; Extended trapezoidal formula; Mathematical analysis; Mathematics; Modified Simpson rule; Newmark method; P-stability; Partial differential equations; Sciences and techniques of general use; Second-order nonlinear hyperbolic equations; Sine-Gordon equation; P-stability; Newmark method; Modified Simpson rule; Sine-Gordon equation; Second-order nonlinear hyperbolic equations; Extended one-step time integration schemes; Extended trapezoidal formula; Second order equation; Neumann problem; Integration; Stability; Initial condition; Differential equation; One step method; Trapezoidal rule; Nonlinear problems; Boundary condition; Jacobi matrix; Partial differential equation; Non linear equation; Simpson ruel; Hyperbolic equation; Pentadiagonal matrix; Dirichlet problem; Tridiagonal matrix; Sine Gordon equation
• ISSN: 0895-7177; 1872-9479
• DOI: 10.1016/S0895-7177(00)00253-3

We present a class of extended one-step time integration schemes for the integration of second-order nonlinear hyperbolic equations u tt = c 2 u xx + p( x,t,u), subject to initial conditions and boundary conditions of Dirichlet type or of Neumann type. We obtain one-step time integration schemes of orders two, three, and four; the schemes are unconditionally stable. For nonlinear problems, the second- and the third-order schemes have tridiagonal Jacobians, and the fourth-order schemes have pentadiagonal Jacobians. The accuracy and stability of the obtained schemes is illustrated computationally by considering numerical examples, including the sine-Gordon equation.