Multistep Scheme Research Articles

Convergence analysis of exponential time differencing scheme for the nonlocal Cahn–Hilliard equation

In this paper, we provide a rigorous proof of the convergence for both first-order and second-order exponential time differencing (ETD) schemes applied to the nonlocal Cahn–Hilliard (NCH) equation. The spatial discretization is executed through the Fourier spectral collocation method, whereas the temporal discretization is implemented using ETD-based multistep schemes. The absence of a higher-order diffusion term in the NCH equation creates a significant challenge for the convergence analysis of numerical schemes. To address this issue, we introduce novel error decomposition formulas and utilize higher-order consistency analysis. These techniques allow us to establish the ℓ∞ bound of the numerical solution under certain natural constraints. By treating the numerical solution as a perturbation of the exact solution, we derive optimal convergence rates in ℓ∞(0,T;Hh−1)∩ℓ2(0,T;ℓ2). Furthermore, we conduct several numerical experiments to validate the accuracy and efficiency of the proposed schemes. These experiments include convergence tests and the observation of long-term coarsening dynamics, thereby confirming the robustness and reliability of our approach.

A novel Lax–Wendroff type procedure of two-derivative time-stepping schemes for Euler and Navier–Stokes equations

In this paper, we developed a novel Lax–Wendroff (LW) type procedure of two-derivative time-stepping schemes for the Euler and Navier–Stokes equations. The explicit two-derivative time-stepping schemes, including the two-derivative Runge–Kutta schemes and variable step size two-derivative multistep schemes, are proposed and the optimized time-stepping schemes are determined by the strong stability preserving theory. The novel LW procedure greatly simplifies the intricate symbolic calculations of the original LW procedure, particularly for the Navier–Stokes equations. The novel LW procedure also ensures both the order accuracy and numerical stability compared to the original LW procedure. Moreover, we presented a simplified positivity-preserving limiter for LW procedure, enabling the temporal schemes to handle demanding computations. When employing high-order spatial methods, numerical tests highlighted that two-derivative time-stepping schemes utilizing the novel LW procedure effectively improve the stability and computational efficiency compared to the Runge–Kutta schemes.

The influence of parasitic modes on stable lattice Boltzmann schemes and weakly unstable multi-step Finite Difference schemes

Numerical analysis for linear constant-coefficient multi-step Finite Difference schemes is a longstanding topic, developed approximately fifty years ago. It relies on the stability of the scheme, and thus—within the L2 setting—on the absence of multiple roots of the amplification polynomial on the unit circle. This allows for the decoupling, while discussing the convergence of the method, of the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that the methods can be essentially studied as if they had only one step. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we demonstrate that in the case of “weakly” unstable Finite Difference schemes with multiple roots on the unit circle, although the schemes may remain stable, considering parasitic modes is essential in studying their consistency and, consequently, their convergence. This research was prompted by unexpected numerical results on stable lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Unlike Finite Difference schemes, rigorous numerical analysis for lattice Boltzmann schemes is a contemporary topic with much left for future discoveries. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of fourth-order schemes. However, this assumption proved incorrect for weakly unstable Finite Difference schemes and for stable lattice Boltzmann methods. This borderline scenario underscores that particular care must be adopted for lattice Boltzmann schemes, and the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors concerning Finite Difference schemes. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle nonlinear systems of conservation laws relying on a Jin-Xin approximation and high-order splitting formulæ.

Multi-step Residual Power Series Method for Solving Stiff Systems

In this paper, an efficient algorithm based on the residual power series method (RPSM) is presented to solve stiff systems of Caputo fractional order. We apply the RPSM on subintervals to get approximate solutions of these types of systems. The RPSM has advantages that it is suitable to solve linear and nonlinear systems and it gives high accurate results. Modifying this technique to multi-step RPSM considerably reduces the number of arithmetic operations and so reduces the time, especially when dealing with Stiff systems. Several numerical examples are given to show the efficiency, simplicity and the accuracy of the proposed method. Comparing classical RPSM with the new multi-step scheme shows that multi-step RPSM controls the convergence behaviour of the stiff systems. That is, the comparison reveals that MS-FRPSM reduces both absolute and residual errors. More iterations and a smaller step size lead to higher accuracy. Moreover, in MS-FRPSM, the intervals of convergence for the series solution will increase.

Concentration of 95–97Mo and production of isotopically modified molybdenum in a multi-step scheme of a square cascade

Concentration of 95–97Mo and production of isotopically modified molybdenum in a multi-step scheme of a square cascade

Multi-step Hermite-Birkhoff predictor-corrector schemes

In this study, we introduce a multi-step multi-derivative predictor-corrector time integration scheme analogous to the schemes in Schütz et al. (2022) [13], incorporating a multi-step quadrature rule. We conduct stability analysis up to order eight and optimize the schemes to achieve A(α)-stability for large α. Numerical experiments are performed on ordinary differential equations exhibiting diverse stiffness conditions, as well as on partial differential equations showcasing non-linearity and higher-order terms. Results demonstrate the convergence and flexibility of the proposed schemes across diverse situations.

Multi-step intelligent prediction of shield machine position attitude on the basis of BWO-CNN-LSTM-GRU

Abstract Realizing automatic control of shield machine tunneling attitude is a challenging problem. Realizing multi-step intelligent prediction for attitude and position is an important prerequisite for solving this problem in the tunneling process with complex and varied geological environments. In this paper, a multi-step intelligent predictive scheme based on beluga whale optimization-convolutional neural network-Long Short-term memory-gated recurrent unit (BWO-CNN-LSTM-GRU) is proposed for shield machine position attitude. First, Pearson correlation analysis is utilized to determine the input feature variables from the construction data and temporalize the input features. Subsequently, CNN-LSTM-GRU predictive models are established for the six positional parameters, separately. Among them, CNN performs feature extraction on the input variables, and LSTM-GRU realizes the predictions for the target positional parameters. In the end, the optimization of the convolutional layer dimension, the number of convolutional layers, iterations, the learning rate, the number of neurons in the LSTM layer and GRU layer of each position predictive model is performed on the basis of BWO, separately, and the best hyperparameters found are built into a BWO-CNN-LSTM-GRU position predictive model, which realizes the multi-step intelligent predictions for the shield machine’s position. The proposed approach is examined by utilizing the Beijing Metro Line 10. The results show that the predictive deviation of the position predictive model is within 3 mm, and the positional trajectory points obtained on the basis of the predicted values and the 3D coordinate system are highly coincident with the actual trajectory points. Therefore, the approach provides a more accurate predictive result for shield attitude and position and can provide a decision-making scheme for further realizing the coordinated autonomous control of shield machine.

A Secure and Efficient Multi-Step Multi-Index Image Encryption Scheme Using Hybrid Combination of Substitution, Permutation and Hyperchaotic Structure

A Secure and Efficient Multi-Step Multi-Index Image Encryption Scheme Using Hybrid Combination of Substitution, Permutation and Hyperchaotic Structure

Multi-step scheme and thermal effects of coal smouldering under various oxygen-limited conditions

Coal smouldering fires are global disasters and notoriously challenging to characterize. To better understand combustion and chemistry within these coal fires, TG-DSC experiments were carried out in two different coal samples in nitrogen and oxygen-limited (including ambient air) atmospheres. A mechanism with 8-step reactions simulating coal combustion was proposed, including one drying, two pyrolysis, two oxygen adsorption, and three high-temperature oxidations. The kinetic parameters were optimized via a Genetic Algorithm (GA). It was found that this 8-step mechanism created unnecessary overfitting to the TGA data. Therefore, two alternative schemes were developed with 5- and 4-step, which included and neglected oxygen adsorption, respectively. To compare the two schemes, GA and Gaussian multi-modal fitting were used to quantify the mass change rates in each step and their thermal effects as functions of oxygen concentrations. Moreover, the heat flow rates of coal samples were calculated and compared with measured DSC curves. The results indicated that oxygen adsorption could play a critical role in the transition from water evaporation to high-temperature combustion, and the 5-step scheme could reflect coal's intrinsic oxygen-limited combustion characteristics more accurately. Altogether, this study provides novel insight into coal smouldering under oxygen-limited conditions.

Polymers with one free gallic acid per monomer unit. Controlled synthesis and intrinsic antioxidant/antibacterial properties

The development of bio-based thermoplastics is rapidly growing due to the depletion of fossil fuel reserves but also in the hope of obtaining polymers with original properties. Herein, we report the controlled synthesis of original homopolymers derived from gallic acid, one of the well-known natural polyphenols that exhibits various health-promoting effects. More precisely, acrylate, methacrylate and acrylamide monomers based on a protected gallic acid are firstly obtained by a rethought multistep scheme. Then, a controlled photo-mediated RAFT (Reversible Addition Fragmentation chain Transfer) polymerization is carried out before a final deprotection. By this way, original polymers bearing a free gallic acid as a lateral substituent on each monomer unit are described. The intrinsic antibacterial and antioxidant properties of these polymers are highlighted and compared with those of gallic acid.

A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear Term

A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear Term

Multi-step variant of the parareal algorithm: convergence analysis and numerics

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems involving a multi-step time scheme by the parareal algorithm. The parareal method is based on combining predictions made by a coarse and cheap propagator, with corrections computed with two propagators: the previous coarse and a precise and expensive one used in a parallel way over the time windows. A multi-step time scheme can potentially bring higher approximation orders than plain one-step methods but the initialisation of each time window needs to be appropriately chosen. Our main contribution is the design and analysis of an algorithm adapted to this type of discretisation without being too much intrusive in the coarse or fine propagators. At convergence, the parareal algorithm provides a solution that coincides with the solution of the fine solver. In the classical version of parareal, the local initial condition of each time window is corrected at every iteration. When the fine and/or coarse propagators is a multi-step time scheme, we need to choose a consistent approximation of the solutions involved in the initialisation of the fine solver at each time windows. Otherwise, the initialisation error will prevent the parareal algorithm to converge towards the solution with fine solver’s accuracy. In this paper, we develop a variant of the algorithm that overcome this obstacle. Thanks to this, the parareal algorithm is more coherent with the underlying time scheme and we recover the properties of the original version. We show both theoretically and numerically that the accuracy and convergence of the multi-step variant of parareal algorithm are preserved when we choose carefully the initialisation of each time window.

Bi-objective carbon-efficient distributed flow-shop scheduling with multistep electricity pricing

A bi-objective distributed flow-shop scheduling problem with multistep electricity pricing and carbon emissions is studied. One objective is to minimise the completion time of production, and the other is to minimise the total cost of multistep electricity pricing and carbon emissions. A mixed integer programming model and a two-stage knowledge based cooperative algorithm with a local reinforcement strategy are proposed for the problem. The extensive numerical experiments show that the two-stage algorithm was effective statistical significantly in generating non-dominated solution sets and Pareto frontiers. Simulations on Electricity Prices are applied to examine different multistep electricity pricing schemes, and management implications were drawn for both government and companies.

Explosive blast loading effect on transient mechanical responses of aircraft panels with curvilinear fibers: 3D elasticity approach

In the present paper, for the first time, the dynamic loading effect on transient bending and stress distributions of aircraft nanocomposite sandwich panels including curvilinear fibers are presented and explained. Face sheets of the panel contain curvilinear fibers with an orientation angle varies linearly with respect to the horizontal coordinate. The core layer is made of polymer matrix reinforced by graphene platelets (GPLs) with uniform and random orientation distribution. 3-D elasticity theory is used to formulate the transient problem of the panel. The model of the panel is so general to include different time-dependent loads, especially explosive ones. A three-dimensional layerwise-differential quadrature method (LW-DQM) together with a non-uniform rational B-spline-based multi-step time integration scheme is employed to investigate the transient responses of the sandwich panel with variable stiffness composite laminated face sheets (VSCL-FS) and graphene platelets reinforced porous core (GPLR-PC) under dynamic loading. The convergence behavior of the method is examined numerically and to assure its accuracy, the results in the limit cases are compared with those available in the literature. Finally, through the parametric studies, the effects of core porosity distribution and amounts, GPLs weight fraction and boundary conditions on the transient responses of the sandwich panel with VSCL-FS and GPLR-PC subjected to explosive blast loading are investigated. The results show that the addition of GPLs to the core layer decreases the transverse displacement but increases the peak values of the stress components. Also, the core porosity increases the transverse displacement and the stress components. However, it has not significant effect on the period of oscillation of the field variables under explosive blast loading. On the other hand, the curvature of the fibers does not considerable effect on the plate response and on the period of oscillation of the field variables under explosive blast loading

Arc Discharge Synthesis and Multistep Purification of Multiwall Carbon Nanotubes

This research work describes the cost-effective synthesis and purification of multiwall carbon nanotubes (MWCNTs). Synthesis of CNTs was carried out in distilled water between two electrodes using the electric arc discharge (EAD) method. EAD is a simple and straightforward route in which an electric arc is generated between graphite electrodes through DC power source to produce soot which contains MWCNTs along with impurities. The deposited soot containing MWCNTs was then chipped off and purified. In this case, multistep purification scheme was opted to remove unwanted impurities from produced MWCNTs. Purification route comprised thermal treatment, chemical treatment and a combination of both to yield pure MWCNTs. Thermal treatments were carried out in normal air and under controlled flow of oxygen at different temperatures whereas chemical treatment was performed using acidic solution. Thermo gravimetric analysis (TGA), field emission scanning electron microscopy (FE-SEM) and X-ray diffraction (XRD) were carried out before and after purification treatments to investigate the outcome of employed treatments. Results showed that the thermal or chemical treatment alone is not sufficient to remove impurities from soot. Moreover, the introduction of an oxide group through chemical treatment reduces the oxidization temperature of graphitic particles. It was found that the chemical treatment followed by thermal annealing under the controlled flow of oxygen is the most appropriate method for successful purification of MWCNTs synthesized via EAD method.

Generalized Convergence for Multi-Step Schemes under Weak Conditions

We have developed a local convergence analysis for a general scheme of high-order convergence, aiming to solve equations in Banach spaces. A priori estimates are developed based on the error distances. This way, we know in advance the number of iterations required to reach a predetermined error tolerance. Moreover, a radius of convergence is determined, allowing for a selection of initial points assuring the convergence of the scheme. Furthermore, a neighborhood that contains only one solution to the equation is specified. Notably, we present the generalized convergence of these schemes under weak conditions. Our findings are based on generalized continuity requirements and contain a new semi-local convergence analysis (with a majorizing sequence) not seen in earlier studies based on Taylor series and derivatives which are not present in the scheme. We conclude with a good collection of numerical results derived from applied science problems.

Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor’s series. Theoretical stability and convergence analysis is established in the L2-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.

Multi-step kinetic mechanism coupled with CFD modeling of slow pyrolysis of biomass at different heating rates

This study employed numerical modeling to investigate the reactivity and kinetics involved in the slow pyrolysis of lignocellulosic biomass. The model integrates a biomass multi-step kinetics scheme with heat, momentum, and mass transfer, and it was based on the transport of species and flow in a porous medium approach. To develop the multi-step kinetics model, two samples of biomass, avocado stone (AS) with a high hemicellulose content and α-cellulose (CEL), were subjected to TGA experiments in an inert atmosphere. Furthermore, several experimental data in the literature on the evolution of slow pyrolysis products and data obtained by TGA experiments were considered to refine and validate the proposed kinetic mechanism. The temperature range, from 25 to 700 °C, was explored using different heating rates (10, 20, and 40 °C/min). Experimental results showed that CO and CO2 are the predominant gases during primary devolatilization, whereas H2 and CH4 result from secondary reactions at temperatures above 400 °C. The proposed mechanism involves computational fluid dynamic (CFD) simulations of laboratory-scale biomass pyrolysis, comparing temperature and species concentrations with experimental data. The predicted results for individual non-condensable gas mass yields showed an average relative error of below 6.90 % and 11.59 % for CEL and AS, respectively. In the case of biochar, the error was 6.41 % and 9.74 % for AS and CEL, respectively. The developed kinetic model can be applied to simulate the slow pyrolytic degradation of biomass based on its chemical composition.

High order schemes for gradient flow with respect to a metric

New criteria for energy stability of multi-step, multi-stage, and mixed schemes are introduced in the context of evolution equations that arise as gradient flow with respect to a metric. These criteria are used to exhibit second and third order consistent, energy stable schemes, which are then demonstrated on several partial differential equations that arise as gradient flow with respect to the 2-Wasserstein metric.

Novel Multistep Implicit Iterative Methods for Solving Common Solution Problems with Asymptotically Demicontractive Operators and Applications

It is very meaningful and challenging to efficiently seek common solutions to operator systems (CSOSs), which are widespread in pure and applied mathematics, as well as some closely related optimization problems. The purpose of this paper is to introduce a novel class of multistep implicit iterative algorithms (MSIIAs) for solving general CSOSs. By using Xu’s lemma and Maingé’s fundamental and important results, we first obtain strong convergence theorems for both one-step and multistep implicit iterative schemes for CSOSs, involving asymptotically demicontractive operators. Finally, for the applications and profits of the main results presented in this paper, we give two numerical examples and present an iterative approximation to solve the general common solution to the variational inequalities and operator equations.