A new fast algorithm based on half-step discretization for one space dimensional quasilinear hyperbolic equations

In this article, we describe a new compact three level implicit method of order four in time and space based on half-step discretization for one space dimensional quasilinear hyperbolic equation utt=A(x,t,u)uxx+f(x,t,u,ux,ut) defined in the semi-infinite solution region, where A > 0. We require only nine grid points for the unknown variable u(x, t) and two extra half-step points each for x- and t-variables. The proposed method is directly applicable to wave equation with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. We describe the derivation of the method in detail. The proposed method when applied to damped wave equation is shown to be unconditionally stable. Many benchmark problems are solved to confirm the fourth order convergence of the proposed method.