An unconditionally stable maximum-principle-preserving numerical method for a normalized time-fractional diffusion equation

Xinpei Wu; Ke Zhang; Juho Ma; Junseok Kim; Xinpei Wu; Ke Zhang; Juho Ma; Junseok Kim

doi:10.3934/math.2026922

AIMS Mathematics

2026, Volume 11, Issue 7: 22897-22919. doi: 10.3934/math.2026922

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Research article

An unconditionally stable maximum-principle-preserving numerical method for a normalized time-fractional diffusion equation

1.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
2.
School of Mathematics and Statistics, Heze University, Heze 274015, China
3.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Received: 19 May 2026 Revised: 30 June 2026 Accepted: 16 July 2026 Published: 28 July 2026
MSC : 35R11, 39A14, 80M20

In this paper, we present the mathematical and numerical analysis of a numerical method for a normalized time-fractional diffusion equation. The method is unconditionally stable and preserves the maximum principle. The normalized derivative was defined using a weighting function. The integral of this function is always equal to one for all fractional orders and times. This property makes it possible to compare the diffusion behavior under different fractional orders in a fair way. We considered a recently proposed finite difference scheme with an implicit time discretization for the equation. The resulting algebraic system has a tridiagonal matrix. The Thomas algorithm was applied to solve the system efficiently. We proved that the numerical scheme is unconditionally stable, satisfies the maximum principle, and converges correctly. The convergence order is $ O(\Delta t^{2-\alpha}) $ in time and $ O(h^2) $ in space. Several numerical examples were also presented. The results agree with the theoretical analysis and further validate the influence of different fractional orders on the diffusion behavior.
- unconditionally stable numerical scheme,
- normalized time-fractional derivative,
- convergence analysis
Citation: Xinpei Wu, Ke Zhang, Juho Ma, Junseok Kim. An unconditionally stable maximum-principle-preserving numerical method for a normalized time-fractional diffusion equation[J]. AIMS Mathematics, 2026, 11(7): 22897-22919. doi: 10.3934/math.2026922

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Abstract

In this paper, we present the mathematical and numerical analysis of a numerical method for a normalized time-fractional diffusion equation. The method is unconditionally stable and preserves the maximum principle. The normalized derivative was defined using a weighting function. The integral of this function is always equal to one for all fractional orders and times. This property makes it possible to compare the diffusion behavior under different fractional orders in a fair way. We considered a recently proposed finite difference scheme with an implicit time discretization for the equation. The resulting algebraic system has a tridiagonal matrix. The Thomas algorithm was applied to solve the system efficiently. We proved that the numerical scheme is unconditionally stable, satisfies the maximum principle, and converges correctly. The convergence order is $ O(\Delta t^{2-\alpha}) $ in time and $ O(h^2) $ in space. Several numerical examples were also presented. The results agree with the theoretical analysis and further validate the influence of different fractional orders on the diffusion behavior.

References

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