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Modified Crank–Nicolson Scheme with Richardson Extrapolation for One-Dimensional Heat Equation


In this paper, Modified Crank–Nicolson method is combined with Richardson extrapolation to solve the 1D heat equation. The method is found to be unconditionally stable, consistent and hence the convergence of the method is guaranteed. The method is also found to be second-order convergent both in space and time variables. When combined with the Richardson extrapolation, the order of the method is improved from second-to fourth-order. To validate the proposed method, two model examples are considered and solved for different values of spatial and temporal step lengths.

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Availability of data and materials

All data generated or analyzed during this study are all included.

Code availability

All the numerical results and the simulations in this article are generated by MATLAB codes using R2013a version and are available.


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Correspondence to Feyisa Edosa Merga.

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Merga, F.E., Chemeda, H.M. Modified Crank–Nicolson Scheme with Richardson Extrapolation for One-Dimensional Heat Equation. Iran J Sci Technol Trans Sci 45, 1725–1734 (2021).

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  • Heat equation
  • Modified Crank–Nicolson scheme
  • Order of convergence
  • Richardson extrapolation
  • Unconditionally stable