Linearly implicit one-step time integration scheme for nonlinear hyperbolic equations

We present a linearized linearly implicit version of the well-known (functionally implicit) Newmark method for initial-value problems for second order ordinary differential equations: y″ = p(y); the linearized method has the same local truncation error and stability properties as the Newmark method. We then employ the linearized method to obtain a linearly implicit one-step time integration scheme for nonlinear hyperbolic equations: utt = c2uxx+p(u); the resulting scheme is unconditionally stable and it obviates the need to solve nonlinear systems at each time step of integration. We demonstrate the computational performance of the linearly implicit scheme for nonlinear ordinary differential equations and for nonlinear hyperbolic equations, including the sine-Gordon equation.