An O(k^{2}+kh^{2}+h^{2}) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations ()

Ranjan Kumar Mohanty, Vijay Dahiya

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**DOI: **10.4236/ajcm.2011.11002
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In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0< x <1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

Keywords

Quasi-Linear Parabolic Equation, Implicit Method, Cubic Spline Approximation, Diffusion-Convection Equation, Singular Equation, Burgers’ Equation, Reynolds Number

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R. Mohanty and V. Dahiya, "An O(k^{2}+kh^{2}+h^{2}) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations," *American Journal of Computational Mathematics*, Vol. 1 No. 1, 2011, pp. 11-17. doi: 10.4236/ajcm.2011.11002.

Conflicts of Interest

The authors declare no conflicts of interest.

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