First Difference Research Articles

Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the L 2 L^2 norm of the numerical solution does not increase in time, under the time step condition τ ≤ F ( h / c , d / c 2 ) \tau \le \mathcal {F}(h/c, d/c^2) , with the convection coefficient c c , the diffusion coefficient d d , and the mesh size h h . The function F \mathcal {F} depends on the specific IMEX temporal method, the polynomial degree k k of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes τ ≲ h / c \tau \lesssim h/c in the convection-dominated regime and it becomes τ ≲ d / c 2 \tau \lesssim d/c^2 in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

Characterization of Surface–Borehole Transient Electromagnetic Response in Electrical Anisotropic Media

When encountering sedimentary rocks with obvious laminations or fracture development zones, the conductivity of the conductive medium in different directions will change significantly, and the subsurface medium will exhibit macroscopic conductivity anisotropy. To analyze the impact of electrical anisotropy on the surface–borehole transient electromagnetic exploration method, we used the finite element method to investigate the electrical anisotropy surface–borehole transient electromagnetic three-dimensional (3D) forward algorithm, in which we used a tetrahedral mesh to spatially discretize the time–domain Maxwell equation. Then, we discretized it using the second-order backward Eulerian difference method, and we obtained the fields through the PARDISO solver. The validity and correctness of the algorithm were verified through comparison of a one-dimensional (1D) anisotropic model, a complex three-dimensional (3D) isotropic model, and a three-dimensional (3D) anisotropic half-space model. A typical anisotropic geological model was constructed to analyze the effects of anisotropic strata and anomalies in the different principal axis directions on the surface–borehole transient electromagnetic response. The results show that the response of the anisotropic medium is related to the direction of the transmitting source, and the response pattern is complex and volatile. The electrical anisotropic anomaly does affect the amplitude, which should be given special attention when performing surface–borehole transient electromagnetic inversion interpretation.

Speed-command-independent parameters self-tuning and mismatch compensation for servo speed control

Usually the servo system using the traditional generalized predictive control is limited by the reference command, the expression of which is only a step signal or a ramp signal. Meanwhile, during the identification convergence, the model mismatch composed of the modeling errors and the nonlinear disturbance, often leads to the obvious speed spike and even damping oscillation. To solve the above problems, a speed-command-independent 2 degree-of-freedom (2DOF) N-type PI (NPI) controller based on an improved generalized predictive control (IGPC) is designed, as well as an extended sliding-mode compensator (ESMC) is proposed in this paper. First, to enhance the steady-state performance for tracking speed command, a 2DOF NPI speed controller is utilized to track the arbitrary-order polynomial command. Then, based on the forward N-order difference operator, an extended prediction model, the output of which only contains the speed error, is established to free the IGPC from the speed command. Next, the optimal control law is obtained by the rolling optimization of a reconstructed cost function, and further is simplified to match with the 2DOF NPI controller online. Finally, the ESMC is designed to effectively weaken the speed spike caused by the model mismatch, and a special positive definite function is utilized to ensure its stability. Simulation and experimental results verify the advantages of the proposed scheme.

Darcy–Forchheimer natural convection flow of SWCNTs-EG-based nanofluid along a complex roughened surface

The study of heat transfer in natural convection flow of nanofluid (mixture of SWCNTs and ethylene glycol fluid) through a complex roughened surface saturated in a porous medium is investigated. The governing equations in terms of nonlinear PDEs are derived using Darcy–Forchheimer and Tiwari and Das models and made dimensionless using transformation. A finite difference scheme with backward and central differences approximations is used to conduct numerical solution. Recently, researchers have focused on the use of roughened surfaces or suspension of nanoparticles in base fluid to improve heat transfer rate. The objective of this study is to quantify the increase in heat transfer rate due to complex roughened surface and nanofluid. The results reveal that total heat transfer rate raises higher in nanofluid as compared to largest amplitude of harmonic wave. The absolute value of drag coefficient and rate of total heat transfer grow by increasing the value of Darcy number. Near the leading edge, the total heat transfer rate declines up-to certain node value which is 0.2125 and 0.2071 in the absence and presence of nanofluid and increases away from these node values. Streams slices and isotherm lines expand in case of largest amplitude of harmonic wave and nanofluid.

Yaw Optimisation for Wind Farm Production Maximisation Based on a Dynamic Wake Model

In recent years, a major focus on wind farm wake control is to maximise the production of wind farms. To improve the power generation efficiency of wind farms through wake regulation, this study investigates yaw optimisation for wind farm production maximisation from the perspective of time-varying wakes. To this end, we first deduce a simplified dynamic wake model according to the momentum conservation theory and backward difference method. The accuracy of the proposed model is verified by simulation comparisons. Then, the time lag of wake propagation and its impact on wind farm production maximisation through wake meandering is analysed. On this basis, a yaw optimisation method for increasing wind farm energy capture is presented. This optimisation method uses the proposed dynamic wake model for wind farm prediction. The results indicate that the optimisation period is critical to the effect of the optimisation method on wind farm energy capture.

A novel conformable fractional nonlinear grey multivariable prediction model with marine predator algorithm for time series prediction

To promote the development of multivariate prediction modelling in small sample environments, this paper constructs a new multivariate prediction model named CFDNGBM(r,N) by integrating the marine predator algorithm (MPA), the modified conformable fractional-order accumulation operation (MCFAO) and the grey prediction model. In CFDNGBM(r,N), the MCFAO and time-delay polynomial are used to enhance the model prediction performance, the backward difference operation is used to activate the unbiasedness, and the unbiased regularization algorithm is used to estimate the model parameters. In addition, the MPA is used to optimize the hyperparameters in the model. The experimental results show that the CFDNGBM(r,N) outperforms the 12 benchmark algorithms and all the optimization measures are effective, both of which confirm the effectiveness of the proposed methods.

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier–Stokes equation with periodic boundary conditions (Huang and Shen, SIAM J. Numer. Anal. 59 (2021) 2926–2954) to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solving Navier–Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.

Decentralized linear–quadratic–Gaussian control of networked control systems with asymmetric information

AbstractThe decentralized linear–quadratic–Gaussian (LQG) control problem for networked control systems (NCSs) with asymmetric information is investigated, where controller 1 shares its historical information with controller 2, and not vice versa. The asymmetry of the information structure leads to the coupling between controller 2 and estimator 1, and hence the classical separation principle fails. Through the assumption of linear control strategy, the coupling between controller 2 and estimator 1 (CCE) is decoupled, but the estimation gain is still coupled with the control gain. It is noted that the control gain conforms to the backward Riccati equation while estimation gain abides by the forward equation, which is computationally challenging. Applying the stochastic maximum principle, the solvability of the decentralized LQG control problem is reduced to that of corresponding forward and backward stochastic difference equations (FBSDEs). Further, necessary and sufficient conditions for the solvability of optimal control problem are presented by two Riccati equations, one of which is nonsymmetric. Moreover, a novel iterative forward method is proposed to calculate the coupled backward control gain and forward estimation gain.

A novel Cell-Centered nodal integral method for the Convection-Diffusion equation

The nodal integral methods (NIMs) are coarse mesh methods for solving partial differential equations (PDEs) efficiently and accurately. Here, a new variant of the NIM is proposed for the solution of the two-dimensional convection–diffusion equation. The earlier versions of the NIM for fluid flow problems used surface-averaged variables. In contrast, the current scheme is developed in terms of cell-centered values using different treatments of cell interface conditions. The advantage of this scheme is that coupling with other physics is straightforward. Another distinctive feature of the current approach is that the temporal derivative is explicitly addressed, resulting in a system of differential–algebraic equations (DAEs) of index-1. The system of DAEs is then solved using Backward Difference Formula, which provides higher accuracy in time compared to traditional NIM. A few problems with the known analytical solution are solved to demonstrate the efficacy of the current simplified approach.

Microcanonical Thermodynamics of Small Ideal Gas Systems

We consider the thermal, mechanical, and chemical contact of two subsystems composed of ideal gases, both of which are not in the thermodynamic limit. After contact, the combined system is isolated, and the entropy is determined through the use of its standard connection to the phase space density (PSD), where only those microstates at a given energy value are counted. The various intensive properties of these small systems that follow from a derivative of the PSD, such as the temperature, pressure, and chemical potential (evaluated via a backward difference), while equal when the two subsystems are in equilibrium are nevertheless found not to behave in accordance with what is expected from macroscopic thermodynamics. Instead, it is the entropy, defined from its connection to the PSD, that still controls the behavior of these small (nonextensive) systems. We also analyze the contact of these two subsystems utilizing an alternative entropy definition, through its proposed connection to the phase space volume (PSV), where all microstates at or below a given energy value are counted. We show that certain key properties of these small systems obtained with the PSV either do not become equal or do not consistently describe the two subsystems when in contact, suggesting that the PSV should not be used for analyzing the behavior of small isolated systems.

General Discretization Method for Enhanced Kinesthetic Haptic Stability.

The stability of haptic simulation systems has been studied for a safer interaction with virtual environments. In this work, the passivity, uncoupled stability, and fidelity of such systems are analyzed when a viscoelastic virtual environment is implemented using a general discretization method that can also represent methods such as backward difference, Tustin, and zero-order-hold. Dimensionless parametrization and rational delay are considered for device independent analysis. Aiming at expanding the virtual environment dynamic range, equationsto find optimum damping values for maximize stiffness are derived and it is shown that by tuning the parameters for a customized discretization method, the virtual environment dynamic range will supersede the ranges offered by methods such as backward difference, Tustin and zero-order-hold. It is also shown that minimum time delay is required for stable Tustin implementation and that specific delay ranges must be avoided. The proposed discretization method is numerically and experimentally evaluated.

Suboptimal Linear Output Feedback Control of Discrete-Time Systems With Multiplicative Noises

This brief considers the fundamental problem of output feedback control for a class of discrete-time systems with multiplicative noises. It is well-known that the classical separation principle fails due to the existence of multiplicative noises in state and control variables. Thus, the traditional optimal solutions for the controller and estimator can not be obtained. Different from the previous methods of enforced separation principle, the explicit expression of the suboptimal linear controller is derived by using the solution of the forward and backward stochastic difference equations (FBSDEs) and the linear optimal estimator. However, it leads to the other mathematical difficult problem that the control gain and the estimator gain are coupled with each other. Specifically, the control gain obeys the backward Riccati equation and the estimator gain satisfies the forward Riccati equation such that the solutions can not be acquired simultaneously. A novel method of iterative solutions to the coupled backward and forward Riccati equations (CBFREs) is proposed. Numerical examples are illustrated to show that the proposed algorithms have better performance than the previous methods.

Fractional-Step Method with Interpolation for Solving a System of First-Order 2D Hyperbolic Delay Differential Equations

In this article, we consider a delayed system of first-order hyperbolic differential equations. The presence of the delay term in first-order hyperbolic delay differential equations poses significant challenges in both analysis and numerical solutions. The delay term also makes it more difficult to use standard numerical methods for solving differential equations, as these methods often require that the differential equation be evaluated at the current time step. To overcome these challenges, specialized numerical methods and analytical techniques have been developed for solving first-order hyperbolic delay differential equations. We investigated and presented analytical results, such as the maximum principle and stability results. The propagation of discontinuities in the solution was also discussed, providing a framework for understanding its behavior. We presented a fractional-step method using a backward finite difference scheme and showed that the scheme is almost first-order convergent in space and time through the derivation of the error estimate. Additionally, we demonstrated an application of the proposed method to the problem of variable delay differential equations. We demonstrated the practical application of the proposed method to solving variable delay differential equations. The proposed algorithm is based on a numerical approximation method that utilizes a finite difference scheme to discretize the differential equation. We validated our theoretical results through numerical experiments.

Multiphase Flowrate Measurement With Multimodal Sensors and Temporal Convolutional Network

Accurate multiphase flow measurement is vital in monitoring and optimizing various production processes. Deep learning has as of late arose as a promising approach for assessing multiphase flowrate dependent on various customary flow meters. In this paper, we propose a multi-modal sensor and Temporal Convolution Network (TCN) based method to predict the volumetric flowrate of oil/gas two-phase flows. The volumetric flowrates of the liquid and gas phase vary from 0.96 - 6.13 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{m}^{{3}}$ </tex-math></inline-formula> /h and 5.5 - 121.2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{m}^{{3}}$ </tex-math></inline-formula> /h, respectively. The multi-modal sequential sensing data are simultaneously collected from a Venturi tube and a dual-plane Electrical Capacitance Tomography (ECT) sensor in a pilot-scale multiphase phase flow facility. The reference data are derived from the single-phase flowmeters. Z-score and First-Difference (FD) data pre-processing methods are employed to manipulate the collected instantaneous time series multi-modal sensing data. The pre-processed data are utilized for training the TCN model. Experimental results reveal that the TCN model can effectively predict the multiphase flowrate based on the multi-modal sensing data. The results provide guidance on data pre-processing methods for multiphase flowrate estimation and demonstrate the effectiveness of combining multi-modal sensors and TCN for multiphase flowrate prediction under complex flow conditions.

Mathematical modeling of fluid flow and pollutant transport in a homogeneous porous medium in the presence of plate stacks

This study aims at investigating the numerical analysis of pollutant transport in homogeneous porous media with solid plate stacks. The investigation was performed for solid/impervious objects of the same size placed in homogeneous porous media. The pollutant transport equation (i.e., steady-state and time dependent advection-dispersion) chosen in mathematical modeling. Three cases arise on the basis of dispersion coefficients: (a) when dispersion is uniformly constant, (b) when dispersion depends upon magnitude of the velocity, and (c) when dispersion depends upon magnitude of the velocity and directional dispersivities, all these are discussed in detail. Generally, analytical solution of such problems doesn't exist, so all the work is done numerically. The governing partial differential equation of pollutant concentration is approximated by using finite difference technique. Central, one-sided, backward and forward finite difference formulae of the same order are used to discretize the domain.Simulations of velocity potential and stream function are approximated by Matlab software. Then equipotential lines and streamlines are visualized in the form of contours. Both, velocity potential and stream function are harmonic and satisfy Laplace's equation. Fluid flow lines and pollutant concentration are represented graphically for several parameters involved in the study.It is found that entrance/exit length, shape, hydraulic conductivity, the number and position of impervious objects affect the fluid flow and pollutant transport. However, there is no significant affect of heated objects on pollutant transport. Moreover, advection and dispersion depend upon permeability of porous media and properties of solid matrix.To authenticate the Matlab scheme of finite difference, it is verified that fluid as well as pollutant fluxes (in and out) are equal. Moreover, time-dependent problem converges to steady-state form after very long time.For monitoring or forecasting the build up of contamination concentration, the pollutant transport model is considerable. As this model is affected by different parameters which are discussed above, can helps to overcome the pollutant accumulation. The solid object is main key to lessen the contamination in the underground. If the entrance or leakage point of the domain is blocked by impermeable object or filled the vertical column with material of low hydraulic conductivity it ultimately slows down or even refrains the pollutant particles to pass through. The pollutant concentration is also minimized by injecting the bioremedial agents with the help of treatment columns.

Feedback Nash Equilibrium With Packet Dropouts in Networked Control Systems

This brief is concerned with the feedback Nash equilibrium for networked control systems (NCSs) with packet dropouts. The packet dropouts occur when the state information is transformed via a wireless network, causing the fall that the state information is unavailable to design the feedback Nash equilibrium. The main contribution of this brief is that the necessary and sufficient condition for the existence of feedback Nash equilibrium is presented. The key is that by decoupling solving the forward and backward difference equations (FBDEs), the analytical solutions of the two controllers are given in terms of the two coupled Riccati equations.

Adaptive Finite Element Simulation of Double-Diffusive Convection

Double-diffusive convection plays an important role in many physical phenomena of practical importance. However, the numerical simulation of these phenomena is challenging since fine meshes are often required to capture the flow physics. Hence, several different numerical methods have been employed in the past. This work reports the development and application of an adaptive finite element method for the simulation of these phenomena, thereby avoiding the need for the use of very fine meshes over the whole domain. The weak formulation of the conservation equations for mass, momentum, energy and species concentration is used. The Boussinesq approximation relates the density of the fluid to the temperature and/or the species concentration. A second-order backward difference method is used for time discretization and the Galerkin method is employed for spatial discretization. Both adaptive time step and grid refinement techniques are employed, and the code is parallelized using MPI. Three different stabilization methods of the convective-diffusion equations are compared; namely, the streamline upwind Petrov–Galerkin (SUPG) method, and two modified methods aimed at diminishing spurious oscillations that include an artificial diffusion term. This diffusion term may be either isotropic or orthogonal to the streamlines. The addition of artificial isotropic diffusion to the SUPG method provides enhanced stability. The method is applied to double-diffusive finger convection in a sucrose-salt aqueous mixture and a stratified salt solution heated from below. The method accurately reproduces the experimentally observed temporal evolution of the salt fingers in the former case and the location of the interfaces between convective and non-convective zones in the latter.

A fast BDF2 Galerkin finite element method for the one-dimensional time-dependent Schrödinger equation with artificial boundary conditions

In this paper, we propose an efficient numerical scheme with linear complexity for the one-dimensional time-dependent Schrödinger equation on unbounded domains. The artificial boundary method is used to address the unboundedness of the domain. By applying the two-step backward difference formula for time discretization and performing the Z-transform, we derive an exact semi-discrete artificial boundary condition of the Dirichlet-to-Neumann type. To expedite the discrete temporal convolution involved in the exact semi-discrete artificial boundary conditions, we design a fast algorithm based on the best relative Chebyshev approximation of the square-root function. The Galerkin finite element method is used for spatial discretization. By introducing a constant damping term to the original Schrödinger equation, we present a complete error estimate for the fully discrete problem. Several numerical examples are provided to demonstrate the accuracy and efficiency of the proposed numerical scheme.

Assessment of high‐order IMEX methods for incompressible flow

SummaryThis paper investigates the competitiveness of semi‐implicit Runge‐Kutta (RK) and spectral deferred correction (SDC) time‐integration methods up to order six for incompressible Navier‐Stokes problems in conjunction with a high‐order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight‐forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time‐dependent boundary conditions, two third‐order RK methods are identified that perform well in all test cases and clearly surpass all second‐order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca .

The long time error estimates for the second order backward difference approximation to sub-diffusion equations with boundary time delay and feedback gain

We consider time fractional diffusion initial boundary value problem with boundary time delay and feedback gain involving two Caputo fractional derivatives in time. When both the time delay and feedback gain are taken as appropriately small such that the characteristic equation has no solution in the location of the right half plane of the imaginary axis, we derive the solution’s existence, uniqueness and representation. The second-order backward difference time discrete schemes are investigated in time discretization. The lt∞(0,∞;L2) error analysis of the numerical schemes is derived when both the time delay and feedback gain are appropriately small such that the characteristic equation of the discrete problem has no solution in a strip of the right half plane of the imaginary axis, and the order of boundary Caputo fractional derivative is equal to the order or half order of interior domain Caputo fractional derivative. The numerical examples illustrate its accuracy, efficiency and robustness, and are consistent with the theoretical results.