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In this work, we are concerned with the stability and convergence analysis of the second-order backward difference formula (BDF2) with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r k := τ k / τ k −1 < 3.561. Moreover, with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms, the L 2 norm stability and rigorous error estimates are established, under the same step-ratio constraint that ensures the energy stability, i.e., 0 < r k < 3.561. This is known to be the best result in the literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose temporal derivatives are singular at t = 0. The main result is a type of fractional Grönwall inequality and we illustrate its use by outlining some stability and convergence estimates of schemes for fractional reaction-subdiffusion problems. This approach extends earlier work that used the familiar LI approximation to the Caputo fractional derivative, and will facilitate the analysis of higher order and linearized fast schemes.

In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction ${r_k}\\,: = \\,{\\tau _k}/{\\tau _{k - 1}}\\, < \\,\\left( {3\\, + \\,\\sqrt {17} } \\right)/2\\, \\approx \\,3.561$. Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum bound principle of the BDF2 scheme under the time-step ratio restriction rk < 1 + √2 ≈ 2.414 and a practical time-step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.

Peripheral and central sensitizations of the trigeminal nervous system are the main mechanisms to promote the development and maintenance of chronic orofacial pain characterized by allodynia, hyperalgesia, and ectopic pain after trigeminal nerve injury or inflammation. Although the pathomechanisms of chronic orofacial pain are complex and not well known, sufficient clinical and preclinical evidence supports the contribution of the N-methyl-D-aspartate receptors (NMDARs, a subclass of ionotropic glutamate receptors) to the trigeminal nociceptive signal processing pathway under various pathological conditions. NMDARs not only have been implicated as a potential mediator of pain-related neuroplasticity in the peripheral nervous system (PNS) but also mediate excitatory synaptic transmission and synaptic plasticity in the central nervous system (CNS). In this review, we focus on the pivotal roles and mechanisms of NMDARs in the trigeminal nervous system under orofacial neuropathic and inflammatory pain. In particular, we summarize the types, components, and distribution of NMDARs in the trigeminal nervous system. Besides, we discuss the regulatory roles of neuron-nonneuronal cell/neuron-neuron communication mediated by NMDARs in the peripheral mechanisms of chronic orofacial pain following neuropathic injury and inflammation. Furthermore, we review the functional roles and mechanisms of NMDARs in the ascending and descending circuits under orofacial neuropathic and inflammatory pain conditions, which contribute to the central sensitization. These findings are not only relevant to understanding the underlying mechanisms, but also shed new light on the targeted therapy of chronic orofacial pain.

Stability and convergence of the L1 formula on nonuniform time grids are studied for solving linear react ion-sub diffusion equations with the Caputo derivative. A discrete fractional Gronwall inequality is developed for the nonuniform L1 formula by introducing a discrete convolution kernel of Riemann-Liouville fractional integral. To simplify the consistency analysis of the nonuniform L1 formula, we bound the local truncation error in a discrete convolution form and consider a global convolution error involving the discrete Riemann-Liouville integral kernel. With the help of discrete fractional Gronwall inequality and global consistency error analysis, a sharp error estimate reflecting the regularity of solution is obtained for a simple L1 scheme. Numerical examples are provided to verify the sharpness of the error analysis.

A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost O ( M N 2 ) and storage O ( MN ) for the standard L1 formula to O ( M N log N ) and O ( M log N ) , respectively, for M grid points in space and N levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time t = 0 , and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a recently developed discrete fractional Grönwall inequality, a global consistency analysis and a discrete H 2 energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.

We build an asymptotically compatible energy of the variable-step L2- 1 σ scheme for the time-fractional Allen–Cahn model with the Caputo’s fractional derivative of order α ∈ ( 0 , 1 ) , under a weak step-ratio constraint τ k / τ k - 1 ≥ r ⋆ ( α ) for k ≥ 2 , where τ k is the k -th time-step size and r ⋆ ( α ) ∈ ( 0.3865 , 0.4037 ) for α ∈ ( 0 , 1 ) . It provides a positive answer to the open problem in Liao et al. (J Comput Phys 414:109473, 2020), and, to the best of our knowledge, it is the first second-order nonuniform time-stepping scheme to preserve both the maximum bound principle and the energy dissipation law of time-fractional Allen–Cahn model. The compatible discrete energy is constructed via a novel discrete gradient structure of the second-order L2- 1 σ formula by a local-nonlocal splitting technique. It splits the discrete fractional derivative into two parts: one is a local term analogue to the trapezoid rule of the first derivative and the other is a nonlocal summation analogue to the L1 formula of Caputo derivative. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our scheme and the asymptotic properties of the associated modified energy.

Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo’s derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time steps are always incorporated into our approaches so that a adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Grönwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.

In consideration of the initial singularity of the solution, a temporally second-order fast compact difference scheme with unequal time-steps is presented and analyzed for simulating the subdiffusion problems in several spatial dimensions. On the basis of sum-of-exponentials technique, a fast Alikhanov formula is derived on general nonuniform meshes to approximate the Caputo’s time derivative. Meanwhile, the spatial derivatives are approximated by the fourth-order compact difference operator, which can be implemented by a fast discrete sine transform via the FFT algorithm. So the proposed algorithm is computationally efficient with the computational cost about O ( M N log M log N ) and the storage requirement O ( M log N ) , where M and N are the total numbers of grids in space and time, respectively. With the aids of discrete fractional Grönwall inequality and global consistency analysis, the unconditional stability and sharp H 1 -norm error estimate reflecting the regularity of solution are established rigorously by the discrete energy approach. Three numerical experiments are included to confirm the sharpness of our analysis and the effectiveness of our fast algorithm.

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn–Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise L 2 norm error estimate is established at the first time under an updated step-ratio restriction 0 < r k : = τ k / τ k - 1 ≤ r user , where r user can be chosen by the user such that r user < 4.864 . The stabilized convex-splitting BDF2 scheme is shown to be mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if 0 < r k ≤ r user , such that it is mesh-robustly stable in an energy norm. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is applied to efficiently capture the multi-scale behaviors and to accelerate the long-time simulation approaching the steady state.