Splitting Method Research Articles

Positivity preserving and unconditionally stable numerical scheme for the three-dimensional modified Fisher–Kolmogorov–Petrovsky–Piskunov equation

This paper introduces a numerical approach for the practical solution of the modified Fisher–Kolmogorov–Petrovsky–Piskunov equation that describes population dynamics. The diffusion term and nonlinear term is based on the operator splitting method and interpolation method, respectively. The analytic proof of the discrete maximum principle and positivity preserving for the numerical algorithm is demonstrated. Numerical solution calculated using the proposed method remains stable without blowing up, which implies that the proposed method is unconditionally stable. Numerical studies show that the proposed method is second-order convergence in space and first-order convergence in time. The performance and applicability of the proposed scheme are studied through various computational tests that present the effects of model parameters and evolution dynamics.

Parallel inertial forward–backward splitting methods for solving variational inequality problems with variational inclusion constraints

The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm.

Predicting the performance of lithium adsorption and recovery from unconventional water sources with machine learning

Selective lithium (Li) recovery from unconventional water sources (UWS) (e.g., shale gas waters, geothermal brines, and rejected seawater desalination brines) using inorganic lithium-ion sieve (LIS) materials can address Li supply shortages and distribution issues. However, the development of high-performance LIS materials and the optimization of recovery-related operating parameters are hampered by the variety of production methods, intricate procedures, and experimental expenses. Machine learning (ML) techniques offer potential solutions for enhancing LIS material development. We collected literature data on Li adsorption, categorizing 16 parameters into adsorbent parameters, operating parameters, and solution components. Three tree-based algorithms—Random Forest (RF), Gradient Boosting Decision Trees (GBDT), and Extreme Gradient Boosting (XGBoost)—were used to evaluate the impact of these parameters on lithium adsorption. The grouped random splitting method limited data leakage and mitigated overfitting. XGBoost demonstrated the best performance, with an R² of 0.98 and a root-mean-squared error (RMSE) of 1.72. The SHAP values highlighted that operating parameters were the most influential, followed by adsorbent parameters and coexisting ion concentrations. Therefore, focusing on optimizing operating parameters or making targeted improvements on LIS based on operating conditions will enhance LIS performances in UWS. These insights are crucial for optimizing Li adsorption processes and designing effective inorganic LIS materials.

Space–time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation

The space–time adaptive ADER–DG finite element method with LST–DG predictor and a posteriori sub–cell ADER–WENO finite–volume limiting was used for simulation of multidimensional reacting flows with detonation waves. The presented numerical method does not use any ideas of splitting or fractional time steps methods. The modification of the LST–DG predictor has been developed, based on a local partition of the time step in cells in which strong reactivity of the medium is observed. This approach made it possible to obtain solutions to classical problems of flows with detonation waves and strong stiffness, without significantly decreasing the time step. The results obtained show the very high applicability and efficiency of using the ADER–DG–PN method with a posteriori sub–cell limiting for simulating reactive flows with detonation waves. The numerical solution shows the correct formation and propagation of ZND detonation waves. The structure of detonation waves is resolved by this numerical method with subcell resolution even on coarse spatial meshes. The smooth components of the numerical solution are correctly and very accurately reproduced by the numerical method. Non–physical artifacts of the numerical solution, typical for problems with detonation waves, such as the propagation of non–physical shock waves and weak detonation fronts ahead of the main detonation front, did not arise in the results obtained. The results of simulating rather complex problems associated with the propagation of detonation waves in significantly inhomogeneous domains are presented, which show that all the main features of detonation flows are correctly reproduced by this numerical method. It can be concluded that the space–time adaptive ADER–DG–PN method with LST–DG predictor and a posteriori sub–cell ADER–WENO finite–volume limiting is perfectly applicable to simulating fairly complex reacting flows with detonation waves.

Interaction of liquid crystals with a rigid body

This article investigates the interaction of nematic liquid crystals modeled by a simplified Ericksen-Leslie model with a rigid body. It is shown that this problem is locally strongly well-posed, and that it also admits a unique, global strong solution for initial data close to constant equilibria. The proof of the global strong solution relies on a new splitting method for the director in a mean value zero and average part.

Mechanical and fracture properties of concrete with recycled concrete aggregates treated with acids and addition of aluminium sulphate

The use of recycled aggregates, particularly concrete waste aggregate, has gained increasing popularity and has become current practice in day-to-day applications in many countries. Enhancing the physical and morphological characteristics of this type of aggregate is crucial to promote its widespread adoption. This study investigates the mechanical and fracture properties of concrete incorporating 100 % recycled concrete coarse aggregate (RCA), both untreated and treated with acid solutions of hydrochloric acid (HCl) and sulphuric acid (H2SO4), and addition to the addition of aluminium sulphate (AS). X-ray diffraction (XRD) and X-ray fluorescence (XRF) techniques were employed to characterize the cement and aluminium sulphate (AS). The performance of all mixes was evaluated in terms of mechanical properties, including compressive strength, tensile strength, and secant modulus of elasticity. Experimental fracture tests were conducted using the wedge splitting method on notched specimens aged for 28 days. The results demonstrate an improvement in the mechanical behaviour and fracture resistance of mixes incorporating RCA treated with acid solutions at concentrations of 0.3 M, 1 M, and 3 M for HCl, as well as at a concentration of 0.3 M for H2SO4. However, a decrease in properties was observed at higher concentrations of H2SO4 (1 M and 3 M). Furthermore, the addition of AS also resulted in improvements compared to mixes containing RCA.

Numerical Simulation of Non-Darcy Flow in Naturally Fractured Tight Gas Reservoirs for Enhanced Gas Recovery

In this work, we analyze non-Darcy two-component single-phase isothermal flow in naturally fractured tight gas reservoirs. The model is applied in a scenario of enhanced gas recovery (EGR) with the possibility of carbon dioxide storage. The properties of the gases are obtained via the Peng–Robinson equation of state. The finite volume method is used to solve the governing partial differential equations. This process leads to two subsystems of algebraic equations, which, after linearization and use of an operator splitting method, are solved by the conjugate gradient (CG) and biconjugate gradient stabilized (BiCGSTAB) methods for determining the pressure and fraction molar, respectively. We include inertial effects using the Barree and Conway model and gas slippage via a more recent model than Klinkenberg’s, and we use a simplified model for the effects of effective stress. We also utilize a mesh refinement technique to represent the discrete fractures. Finally, several simulations show the influence of inertial, slippage and stress effects on production in fractured tight gas reservoirs.

Energy-stable auxiliary variable viscosity splitting (AVVS) method for the incompressible Navier–Stokes equations and turbidity current system

In this work, we develop a novel energy-stable linear approach, which we name as auxiliary variable viscosity splitting (AVVS) method, to efficiently solve the incompressible fluid flows. Different from the projection-type methods with pressure correction, the AVVS method adopts the viscosity splitting strategy to split the original momentum equation into an intermediate momentum equation without divergence-free constraint and an advection-free momentum equation. A time-dependent auxiliary variable which has exact value 1 is introduced to construct a supplementary equation. The new model not only inherits the same dynamics of original incompressible Navier–Stokes equations, but also facilitates us to design linearly decoupled and energy-stable time-marching scheme. Comparing with the conventional projection-type schemes, the present method leads to an energy dissipation law with respect to kinetic energy instead of a modified energy including velocity and pressure gradient. In each time step, only two parabolic equations with constant coefficients and one Poisson equation need to be solved. Therefore, the numerical implementation is highly efficient. Moreover, the proposed AVVS method can be directly extended to construct linear, decoupled, and energy-stable scheme for the turbidity current system with slight modifications on the right-hand side of supplementary equation. Extensive numerical experiments are implemented to validate the accuracy, energy stability, and capability in complex fluid simulations.

Reinforcement learning for distributed hybrid flowshop scheduling problem with variable task splitting towards mass personalized manufacturing

Mass personalization manufacturing (MPM), an emerging production pattern, aims to improve enterprise profit in modern industries. However, the processing of heterogeneous orders from the consumers complicates such production scheduling problem. In addition, different scale tasks should adopt different splitting strategies in practical manufacturing, which makes the task splitting method more worthy of investigation. Towards MPM, this paper presents a distributed hybrid flowshop scheduling problem with variable task splitting (DHFSP-VTS) to minimize the makespan and total energy consumption simultaneously. Meanwhile, the VTS allows the tasks to be split into different sublots so they can save setup and transfer time. To solve these problems, we present an order modularization processing method that can categorize multiple types of orders into specific generation tasks, and a highly effective reinforcement learning-multiple objective evolutionary algorithm based on decomposition (RL-MOEA/D) is designed. In RL-MOEA/D, there are three features: (1) three initial rules are used for initialization based on the current splitting scheme that can increase the diversity of solutions; (2) the reinforcement learning agent uses the Q-learning mechanism to dynamically select the scheme of task splitting as action; (3) a neighborhood search strategy improves the exploitation ability and expand the solution space. To verify the effectiveness of RL-MOEA/D, the MOEA/Ds based on four splitting schemes and four RL combined meta-heuristics are compared on 18 instances. The results show that RL-MOEA/D can obtain the best optimization and stability of all the other comparison algorithms. Therefore, it’s a new technique to solve DHFSP with large-scale tasks, especially for the problem of MPM.

Rational synthesis of highly efficient dual–Z–scheme InVO4/FeVO4/Ex–CQDs–g–C3N4 heterojunction for photo(electro)chemical water splitting and pollutant removal applications

BackgroundThe ever–growing concern for environmental pollution and the need for clean energy sources have driven research toward sustainable technologies. Semiconductor photocatalysis has emerged as a promising approach for both environmental remediation and clean energy generation due to its efficiency and environment–friendly nature. This work focuses on developing a novel photocatalyst capable of addressing two crucial environmental challenges: Cr(VI) removal and water splitting for clean hydrogen production. MethodsThis study presents the development of a dual–Z–scheme heterojunction photo(electro)catalyst based on a combination of metal vanadates (FeVO4 and InVO4) and ultrasound–exfoliated carbon–rich graphitic carbon nitride (Ex–C–g–CN), denoted as IVO/FVO/Ex–C–g–CN. The synthesized nanocomposite was thoroughly characterized using various spectroscopic and microscopic techniques (such as XRD, XPS, UV–DRS, FESEM, EDX, and PL) to understand its material properties and structure. These techniques are crucial for elucidating the relationship between the composition of the material and its photocatalytic performance. Significant findingsThe key innovation of this work lies in the design of the dual–Z–scheme heterojunction within the IVO/FVO/Ex–C–g–CN photo(electro)catalyst. This design fosters efficient separation of photogenerated charges, a critical factor for enhancing photocatalytic activity. The effectiveness of this approach is evident in the achieved removal efficiency of 97.17 % for 100 ppm Cr(VI) within just 60 min of visible light irradiation. This demonstrates the superior ability of the developed photocatalyst to address chromium contamination. Furthermore, the photocatalyst exhibits a remarkable photocurrent of 3.16 mA and a low onset potential of 112 mV for the photoelectrochemical oxygen evolution (OER) reaction. These findings highlight the potential of this material for solar–driven water splitting, a clean and sustainable method for hydrogen production. Additionally, the IVO/FVO/Ex–C–g–CN composite demonstrates excellent recyclability, maintaining high Cr(VI) removal efficiency over multiple cycles, indicating its reusability and cost-effectiveness. Overall, the exceptional photo(electro)catalytic performance of the IVO/FVO/Ex–C–g–CN dual–Z–scheme heterojunction positions it as a promising candidate for tackling environmental pollution and generating clean energy.

Convergence analysis of an efficient scheme for the steady state second grade fluid model

We are interested in studying the stationary second grade fluid model in a bounded domain in R2. To approximate the solution of the continuous model, we propose a fully decoupled numerical scheme based on a splitting method combined with the use of the Grad–Div operator. This approach allows the complete decoupling of the three variables: velocity, pressure and vorticity. Each variable is computed using an iterative procedure, with the pressure step involving a simple L2-projection.We provide a proof of the convergence of the scheme to the continuous problem under smallness assumptions on the data. This theoretical analysis ensures the reliability of our method in approximating the behavior of the stationary second grade fluid model.Finally, we present several numerical tests to validate our approach. These tests illustrate the effectiveness and efficiency of our scheme in various scenarios, highlighting its potential applicability to a wide range of problems involving second grade fluids.

A periodic split attractor reconstruction method facilitates cardiovascular signal diagnoses and obstructive sleep apnea syndrome monitoring

Electrocardiogram (ECG) is a powerful tool to detect cardiovascular diseases (CVDs) and health conditions. We proposed a new method for evaluating ECG for efficient medical diagnosis in daily life. By splitting the signal according to the cardiac activity cycle, the periodic split attractor reconstruction (PSAR) method is proposed with time embedding, including three types of splitting methods to show its chaotic domain characteristics. We merged the CVDs dataset and the obstructive sleep apnea syndrome (OSAS) first-lead ECG signal dataset to validate the performance of PSAR for diagnosis and health monitoring using PSAR density maps as SE-ResNet input features. PSAR under 3 split methods showed different sensitivities for different CVDs. While in OSAS monitoring, PSAR showed good ability to recognize sleep abnormalities.

Reliability of In-band and Broadband Spectral Index Measurement: Systematic Study of the Effect of Signal-to-noise Ratio for uGMRT Data

Low-radio-frequency spectral index measurements are a powerful tool for distinguishing between different emission mechanisms and, in turn, understanding the nature of the sources. Besides the standard method of estimating the “broadband” spectral index of sources from observations in two different frequency “bands,” if the observations were made with large instantaneous bandwidth, the “in-band” spectral index can be determined, either using images of emission at multiple frequency ranges within a band or using the novel Multi Term-Multi Frequency Synthesis (MT-MFS) imaging algorithm. Here, using simulated upgraded Giant Metrewave Radio Telescope (uGMRT) data, we have systematically studied the reliability of various methods of spectral index estimation for sources with a wide range of signal-to-noise ratios (S/Ns). It is found that for synthetic uGMRT point-source data, the MT-MFS imaging algorithm produces in-band spectral indices for S/N ≲ 100 that have errors ≳0.2, making them unreliable. However, at a similar S/N, the sub-band splitting method produces errors ≲0.2, which are more accurate and unbiased than the in-band spectral indices. The broadband spectral indices produce errors ≲0.2 even for S/N ≳ 15, and hence they are most reliable if there are no higher-order variations in the spectral index. These results may be used to improve the uGMRT observation and data analysis strategies, depending on the brightness of the target source.

A fixed-stress splitting method for nonlinear poroelasticity

A fixed-stress splitting method for nonlinear poroelasticity

A quick probability-oriented introduction to operator splitting methods

The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are provided. The bibliography contains illustrative deterministic examples and a selection of probability-related works.

One-level and two-level operator splitting methods for the unsteady incompressible micropolar fluid equations with double diffusion convection

In this paper, a one-level operator splitting method (OOSM) and a two-level operator splitting method (TOSM) for the two-dimensional or three-dimensional (2D/3D) unsteady incompressible micropolar fluid equations with double diffusion convection (IMFDDC) are proposed and analyzed. Firstly, a OOSM is constructed, which consists of five steps. The projection strategy is adopted to get the values of linear velocity and pressure in the first two steps, then the three elliptic subproblems about the angular velocity, the temperature, and the concentration variables are solved in the last three steps. The original variables with the predefined boundary conditions need to be solved separately in each step. Next, in order to solve the problem efficiently, a TOSM is presented, which only solves the nonlinear equations in the coarse level subspaces and the Oseen type linearized equations in the fine level. The numerical analysis of stability and error estimates of the fully discrete methods show that TOSM can reach the same accuracy as the OOSM with H∼O(h2/3). Numerical examples are provided to confirm the effectiveness and reliability of the proposed methods.

Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: Applications to fluid turbulence and generation of fractional Gaussian fields

Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics was recently proposed that can generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear partial differential equation stirred by an additive random forcing term that is δ-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space that aims to transfer energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We present here simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudospectral simulations. We show that our algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two, and three spatial dimensions, and we display the solutions both in physical and Fourier space. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities. Published by the American Physical Society 2024

Modified general splitting method for the split feasibility problem

Modified general splitting method for the split feasibility problem

Low Mach number approximated linearized perturbed Euler equations-based unified computational flow-induced acoustic model for 2D planar/axisymmetric complex geometry

This article is on proposition of a unified 2D planar and axisymmetric differential formulation—for both fluid dynamics and acoustic governing equations. Further, using hydrodynamic splitting method for computational acoustics, a differential formulation is proposed by performing low Mach number approximation on Linearized Perturbed Euler Equations (LPEE). Finally, using the differential formulations, a unified numerical methodology is proposed for Computational Flow-induced Acoustics (CFiA)—involving complex geometry—on a body-fitted curvilinear grid. A finite volume and finite difference methods-based hybrid method is presented for algebraic formulation in CFiA. Further, solution methodology is presented with a semi-implicit pressure projection method for fluid flow, four step Runge-Kutta method along with sub time-stepping for acoustic perturbations, and a one-way explicitly coupled solution algorithm for the CFiA. A detailed validation study is presented separately for the present in-house computational-acoustic and CFiA solvers—by considering various test problems, involving 2D planar and axisymmetric complex geometry problems. Further, using an order of accuracy study, the present acoustic and CFiA solvers are demonstrated as IV-order and II-order accurate, respectively. Finally, using the CFiA solver, a superior performance is demonstrated for the proposed low Mach number approximated LPEE as compared to the traditional LPEE-based solver. For low Mach number CFiA, involving 2D planar and axisymmetric complex geometries, a novel method is presented here for computationally efficient CFiA development.

Using Adiabatic Energy Splitting To Compute Dexter Energy Transfer Couplings.

Dexter energy transfer and transport (DET) are of broad interest in energy science, and DET rates depend on electronic couplings between donor and acceptor species. DET couplings are challenging to compute since they originate from both one- and two-particle interactions, and the strength of this interaction drops approximately exponentially with donor-acceptor distances. Using adiabatic energy splitting to compute DET couplings has advantages because adiabatic states can be calculated directly using conventional quantum chemical methods. We describe a minimum energy splitting method to compute the DET coupling by altering molecular geometries to drive the systems into a T1/T2 energy quasi-degenerate-activated DA complex. We explore the accuracy of various quantum chemical approaches to calculate the Dexter couplings.