Fourth-order approximations at first time level, linear stability analysis and the numerical solution of multidimensional second-order nonlinear hyperbolic equations in polar coordinates

In this article, three-level implicit difference schemes of O(k4 + k2h2 + h4) where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, are proposed for the numerical solution of one, two and three space-dimensional nonlinear wave equations in polar coordinates subject to appropriate initial and Dirichlet boundary conditions. We also discuss fourth-order approximation at first time level for more general case. We also obtain the stability range of the difference scheme when applied to a test equation: utt = urr + aruu − ar2u + g(r,t), a = 1 and 2 cal examples are provided to demonstrate the required order of convergence of the methods.