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In the paper, a one-dimensional fractional differential equation describing the process of relaxation filtration of liquid in a porous medium is numerically studied. To solve the equation the finite difference method is used. Two difference schemes are constructed, one semi-implicit and the other - fully implicit. Using the von Neumann method, the stability of these schemes with respect to the initial data is studied. It is shown that at the transition from the first time layer to the second one, the semi-implicit scheme is conditionally stable, and the full implicit scheme is unconditionally stable. It is shown that, based on the numerical analyses of the ratio of amplitudes in the Fourier harmonics the stability condition for the transition from the first time layer to the second one ensures the stability in subsequent time layers. The stability regions of the semi-implicit scheme with respect to time step of the grid are determined, the influence of various parameters on the size of the stability regions of the schemes is estimated. The unconditional stability of the full implicit scheme is shown.

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Stability of Finite-Difference Schemes for the Fractional Differential Equation of Fluid Filtration in Porous Media

  • 会议论文

  • 发表日期：

    2025年

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  • B. Kh. Khuzhayorov,
  • K. K. Wiswanathan,
  • T. O. Dzhiyanov

摘要

In the paper, a one-dimensional fractional differential equation describing the process of relaxation filtration of liquid in a porous medium is numerically studied. To solve the equation the finite difference method is used. Two difference schemes are constructed, one semi-implicit and the other - fully implicit. Using the von Neumann method, the stability of these schemes with respect to the initial data is studied. It is shown that at the transition from the first time layer to the second one, the semi-implicit scheme is conditionally stable, and the full implicit scheme is unconditionally stable. It is shown that, based on the numerical analyses of the ratio of amplitudes in the Fourier harmonics the stability condition for the transition from the first time layer to the second one ensures the stability in subsequent time layers. The stability regions of the semi-implicit scheme with respect to time step of the grid are determined, the influence of various parameters on the size of the stability regions of the schemes is estimated. The unconditional stability of the full implicit scheme is shown.