# A New Spline-in-Tension Method of O(\(k^{2}+h^{4})\) Based on Off-step Grid Points for the Solution of 1D Quasi-linear Hyperbolic Partial Differential Equations in Vector Form

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## Abstract

In this paper, we propose a new three level implicit method based on half-step spline in tension method of order two in time and four in space for the solution of one-space dimensional quasi-linear hyperbolic partial differential equation of the form \(w_{tt}=K(x,t,w)w_{xx} + {\varphi }(x,t,w,w_{x},w_{t})\). We describe spline in tension approximations and its properties using two half-step grid points. The new method for one dimensional quasi-linear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function *w*(*x*, *t*) and two half-step points for the known variable ‘*x*’ in *x*-direction. The proposed method when applied to Telegraphic equation is shown to be unconditionally stable. Further, the stability condition for 1-D linear hyperbolic equation with variable coefficients is established. Our method is directly applicable to hyperbolic equations irrespective of the coordinate system which is the main advantage of our work. The proposed method for scalar equation is extended to solve the system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems and numerical computations are provided to demonstrate the effectualness of the method.

## Keywords

Spline in tension approximations Quasi-linear hyperbolic equations Half-step grid points Telegraphic equation Unconditionally stable Maximum absolute errors## Mathematics Subject Classification

65M06 65M12## Notes

### Acknowledgements

The authors thank the reviewers for their valuable suggestions, which have substantially improved the standard of the paper.

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