Abstract
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.
Highlights
For waves where the wavelength is much greater than water depth, they can be modelled by coupled fluid dynamics equations known as the shallow water wave equations
No numerical methods based on compression spline approximations for the solution of one-space dimensional quasilinear hyperbolic equations on a variable mesh are known in the literature
In this paper, using three non-uniform grid points in x-direction and three uniform grid points in t-direction, we discuss a new threelevel implicit method of accuracy two in time and three in space based on spline in compression approximations for the solution of D second order quasilinear hyperbolic equations
Summary
The waves for all these applications are described by solutions to either linear or nonlinear second order hyperbolic partial differential equations There has been much effort to develop convergent numerical methods based on spline approximations for the solution of differential equations. In , Mohanty and Gopal [ ] originally developed a high accuracy numerical method based on cubic spline approximations for the solution of D nonlinear hyperbolic equations.
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