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

We present a class of extended one-step time integration schemes for the integration of second-order nonlinear hyperbolic equations utt = c2uxx + 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.