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A New Two-Level Implicit Scheme for the System of 1D Quasi-Linear Parabolic Partial Differential Equations Using Spline in Compression Approximations

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Abstract

In this article, we proposed a new two-level implicit method of accuracy two in time and four in space based on spline in compression approximations using two half-step points and a central point on a uniform mesh for the numerical solution of the system of 1D quasi-linear parabolic partial differential equations subject to appropriate initial and natural boundary conditions prescribed. The proposed method is derived directly from the continuity condition of the first order derivative of the non-polynomial compression spline function. The stability analysis for a model problem is discussed. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, we have solved generalized Burgers–Huxley equation, generalized Burgers–Fisher equation, coupled Burgers-equations and parabolic equations with singular coefficients. We show that the proposed method enables us to obtain high accurate solution for high Reynolds number.

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Acknowledgements

The authors thank the reviewers for their valuable suggestions, which substantially improved the standard of the paper. This research work is partly supported by CSIR-SRF, Grant no: 09/045(1161)/2012-EMR-I and partly supported by ‘The Department of Science and Technology, Government of India’ under the ‘Mathematical Research Impact Centric Support (MATRICS) Scheme—Grant no: MTR/2017/000163’.

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Authors and Affiliations

  1. Department of Applied Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, 110021, India

    R. K. Mohanty

  2. Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, 110007, India

    Sachin Sharma

  3. Department of Mathematics, Sri Venkateswara College, University of Delhi, New Delhi, 110021, India

    Swarn Singh

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  1. R. K. Mohanty
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  2. Sachin Sharma
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  3. Swarn Singh
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Correspondence to R. K. Mohanty.

Additional information

This work is partially supported by CSIR-SRF, Grant no.: 09/045(1161)/2012-EMR-I, and partially supported by DST-MATRICS, Grant no.: MTR/2017/000163’.

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Mohanty, R.K., Sharma, S. & Singh, S. A New Two-Level Implicit Scheme for the System of 1D Quasi-Linear Parabolic Partial Differential Equations Using Spline in Compression Approximations. Differ Equ Dyn Syst 27, 327–356 (2019). https://doi.org/10.1007/s12591-018-0427-5

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  • DOI: https://doi.org/10.1007/s12591-018-0427-5

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