Abstract

Time-fractional diffusion equations combine a nonlocal memory operator with diffusion and therefore require time-stepping methods that are accurate without losing the qualitative stability of the underlying problem. This paper provides a controlled computational comparison of the classical L1 formula and the second-order L2-1? formula for a Caputo diffusion equation. The study is designed as a baseline for the development of energy-stable high-order schemes. A standard one-dimensional model is discretised in space by centred differences and in time by the two fractional formulas. Accuracy is assessed with the manufactured solution u(x,t)=t³ sin(?x) for fractional orders ?=0.5 and ?=0.7. For ?=0.5, the observed L1 order approaches 1.47 while the L2-1? method gives approximately second-order convergence. For ?=0.7, the corresponding orders are about 1.29 and 2.03. A second experiment considers the unforced problem with u(x,0)=sin(?x) and monitors the discrete L2 energy. Both methods produce monotone energy decay for the tested step sizes, with the L2-1? trajectory closely matching the L1 baseline. The results confirm the expected accuracy advantage of the high-order formula and show why energy diagnostics should accompany convergence tests. They also identify initial regularity and history cost as the principal issues that must be addressed in subsequent high-order scheme development.

Introduction

1. Introduction

Fractional diffusion equations are now a standard mathematical model for transport processes in which the present state depends on the past. In contrast with the classical heat equation, where the first time derivative is local, a time-fractional model contains a history-dependent operator. This feature is useful for anomalous transport, trapping, viscoelastic relaxation and heterogeneous media, but it also changes the numerical problem fundamentally. Every new time level contains information from all earlier levels, and the solution frequently has reduced temporal regularity near t=0. These two properties make accuracy, stability and computational cost inseparable design questions [1–3].

The PhD research framework from which this paper is developed is concerned with energy-stable high-order time-stepping schemes for fractional diffusion equations. That framework identifies four linked tasks: construction of higher-order temporal discretisations, derivation of a discrete energy structure, rigorous stability and error analysis, and numerical comparison against established methods. The present paper addresses the comparison stage first. This is deliberate. A new scheme can only be evaluated meaningfully when its behaviour is measured against reference formulas whose convergence and stability properties are well understood.

The L1 formula remains the most widely used reference method. It is simple, robust and based on piecewise-linear interpolation of the solution inside the Caputo history integral. For a sufficiently smooth solution on a uniform mesh its local consistency leads to a temporal rate close to 2-?, although weak initial singularities can reduce that rate substantially [6,7]. The L2-1? method proposed by Alikhanov [4] uses a higher-order reconstruction and evaluates the derivative at an off-step point tn+? . With ?=1-?/2 , it gives second-order time accuracy under appropriate regularity and has become an important building block for high-order fractional solvers [5,8,9].

A difficulty in discussing 'stability' is that several notions are used in the fractional literature. Norm stability means that numerical errors do not grow uncontrollably. Energy stability is stronger and more structural: the discretisation should preserve, bound or appropriately mimic a continuous dissipation law. For gradient-flow models this issue is explicit, and recent work has shown that positivity or positive-semidefiniteness properties of discrete fractional operators can be used to derive energy laws for L2-type methods [10–13]. For a linear diffusion problem, a practical preliminary diagnostic is the decay of a discrete L2 energy. Such a diagnostic does not replace a theorem, but it can quickly expose a time integrator that produces nonphysical growth.

The purpose of this paper is therefore threefold. First, it states the L1 and L2-1? discretisations in a common notation. Secondly, it quantifies their temporal accuracy on a smooth manufactured solution while keeping the spatial error small. Thirdly, it examines the discrete energy history for the homogeneous problem. The paper does not claim a new fractional formula. Its contribution is a clean, reproducible benchmark and an analytical interpretation that can serve as the control case for subsequent development of corrected, variable-step or genuinely higher-order energy-stable methods.

Conclusion

This paper compared the L1 and L2-1? formulas for a Caputo time-fractional diffusion equation using a common finite-difference framework. On the smooth manufactured solution, L1 reproduced the expected order near 2-?, while L2-1? delivered approximately second-order convergence for both ?=0.5 and ?=0.7. In the homogeneous diffusion test, both methods produced monotone decay of the discrete L2 energy over the computed interval. The L2-1? method therefore provides a strong high-order baseline: it improves accuracy substantially without producing a visible loss of dissipative behaviour in the tested case. The main limitation is equally clear. Smooth-data accuracy and numerical energy decay are not sufficient to establish a general energy-stable high-order theory. Fractional diffusion solutions may be weakly singular at the initial time, and direct history summation is expensive. The next methodological step should combine nonuniform or corrected starting treatment with a discrete energy proof and fast memory evaluation. These requirements are taken up explicitly in the companion paper on graded temporal meshes and weak initial singularities. From a thesis-development perspective, the benchmark also clarifies what would count as a substantive new contribution. Merely increasing the formal order on a smooth example would not be enough. A stronger scheme should retain its advantage for low-regularity data, admit an energy estimate that is stated in a precise discrete norm, and remain computationally viable when the number of time levels is large. Ideally, the proof should identify explicit conditions on coefficients or step ratios rather than relying only on numerical evidence. The computational study should then verify those analytical conditions across several fractional orders and compare the method against both L1 and L2-1? at matched accuracy. This combination of theorem, singular-data robustness and cost-aware validation would connect the numerical experiments directly to the approved research objectives and provide a defensible basis for a later journal paper centred on the genuinely new method. The two benchmark values of ? used here are also useful for later sensitivity analysis. The case ?=0.5 represents stronger memory and a larger separation between L1 and second-order behaviour, whereas ?=0.7 gives a milder but still clearly fractional regime. Adding ? close to one and ? below one-half in future work would test whether a proposed method remains reliable across both weak-memory and strongly singular limits. The same benchmark can then be reused unchanged, which makes cross-method comparisons more transparent and statistically unnecessary because the outputs are deterministic numerical errors.

References

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2026