A Linearly Implicit One-Step Time Integration Scheme For Nonlinear Hyperbolic Equations In Two Space Dimensions

Recently, Chawla and Al-Zanaidi [1] proposed a linearly implicit (LI) one-step scheme for the time integration of second order ordinary differential equations and for nonlinear hyperbolic equations in one space dimension. In the present paper we investigate an application of the LI-scheme in the locally one-dimensional (LOD) mode for the time integration of second order nonlinear hyperbolic equations in two space dimensions: u_{tt} = c^{2}(u_{xx} + u_{yy}) + p(u) . The present linearly implicit scheme in the LOD mode (LI-LOD) is shown to be unconditionally stable. To demonstrate the computational performance of the LI-LOD scheme, and to compare it with the performance of the (implicit) Newmark scheme in the LOD mode (Newmark-LOD), we consider examples of nonlinear hyperbolic equations, including the well known sine-Gordon equation. The present LI-LOD scheme provides an accuracy comparable with the Newmark-LOD scheme, ignoring the Newton iterations required by the Newmark scheme at each time step of integration.

Keywords:

• Hyperbolic Equations
• Nonlinear
• Two Space Dimensions
• One-step Time Integration
• Newmark Scheme
• Linearly Implicit Scheme
• Unconditional Stability
• Sine-Gordon Equation