High accuracy difference schemes for a class of singular three space dimensional hyperbolic equations

For the numerical integration of the system of 3-D nonlinear hyperbolic equations with variable coefficients, we report two three-level implicit difference methods of 0(k 4 + k 2 h 2 + h 4 ) where k and h are grid sizes in time and space directions, respectively. When the coefficients of u xy , u yz and u yz are equal to zero we require only (7+19 + 7) grid points and when the coefficients of u xy , u yz and u zx are not equal to zero and the coefficients of u xx , u yy and u zz are equal we require (19+27+19) grid points. The three-level conditionally stable ADI method of 0 (k4 + k 2 h 2 + h 4 ) for the numerical solution of wave equation in polar coordinates is discussed. Numerical examples are provided to illustrate the methods and their fourth order convergence.