A new discretization method of order four for the numerical solution of one-space dimensional second-order quasi-linear hyperbolic equation

Abstract

Three level-implicit finite difference methods of order four are discussed for the numerical solution of the mildly quasi-linear second-order hyperbolic equation A (x, t, u)u xx + 2B(x, t, u)u xt + C(x, t, u)u tt = ƒ (x, t, u, ux, ut ), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions. A new technique is introduced to obtain the stability range of the wave equation in polar coordinates. Fourth-order approximation at the first time level for a more general case is also discussed. The fourth-order accuracy of the method is demonstrated computationally by four examples.