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In this paper, we present a new three-level implicit method of order two in time and three in space on a non-uniform mesh, based on spline in compression approximation for the numerical solution of 1D quasilinear second order hyperbolic partial differential equations. We also discuss the application of the proposed method to a wave equation with singular coefficients. Stability analysis of a linear scheme and convergence analysis of a general nonlinear scheme are also discussed in this paper. Computational results are given to demonstrate the usefulness of the proposed method.

A new variable mesh method based on non-polynomial spline in compression approximations for 1D quasilinear hyperbolic equations