High accuracy non-polynomial spline in compression method for one-space dimensional quasi-linear hyperbolic equations with significant first order space derivative term

In this paper, we propose a new three-level implicit nine point compact non-polynomial spline in compression finite difference method of order two in time and four in space directions, based on non-polynomial spline approximation in x-direction and central difference approximation in t-direction for the numerical solution of one-space dimensional second order quasi-linear hyperbolic partial differential equations with first order space derivative term. We describe the mathematical details of the method and also discuss how our method is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.