Backward Difference Scheme Research Articles

A novel dynamic predictive control model for variable air volume air conditioning systems using deep learning

Model-based predictive control has proven effective for managing indoor temperature and humidity in variable air volume (VAV) air conditioning systems. However, delays in temperature and humidity adjustments in response to changes in internal and external environmental conditions can impede high-performance operation. This study introduces a dynamic predictive control model for the multi-zone indoor temperature and humidity of VAV systems, designed to predict and control these parameters in real-time. Utilizing the RC network two-node wall structure method and backward finite difference scheme, a multi-zone building model is developed. Coupled with a deep fuzzy cognitive map (DFCM) for temperature and humidity prediction, and controlled by a PID controller, the model dynamically regulates the opening degrees of terminal air dampers and chilled water valves. Implemented on a networked control platform in a large commercial building, the model consistently maintained indoor temperature and humidity within ±0.5 °C and ±0.1 g/(kg dry air) of the target values under varying conditions, demonstrating substantial improvements in stability and operational efficiency. The research enhances the predictability and control of HVAC systems through advanced algorithmic integration, improving real-time indoor climate management in complex building environments. This method offers a practical improvement in HVAC control technologies, which could be beneficial for applications in commercial building management systems, providing a useful tool for supporting energy efficiency and sustainability in modern infrastructures.

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

In this study, numerical solutions of the unsteady convection–diffusion type equations of variable convective coefficients are investigated by two effective techniques alternative to direct boundary element method (BEM). That is, the domain boundary element and the dual reciprocity BEMs with the fundamental solution of convection–diffusion equation are used in space to transform the governing differential equations into equivalent integral equations while a backward finite difference scheme is utilized in time discretization. The fruitfulness of these combined techniques are shown by the implementation of the techniques for some widely investigated fluid dynamics problems. In fact, the lid-driven cavity flow is solved, and precise results are attained as benchmarks for assessing the accuracy of the aforementioned methods. Further, the application of the methods is extended for the natural convection flow which is governed mainly by the convection–diffusion type equations accompanied by the velocity components as variable convective coefficients. The obtained numerical results reveal that the domain BEM which uses the fundamental solution of convection–diffusion equation captures the characteristics and physical advancement of the fluid flow quite well at various combined values of the problem physical parameters.

3-D forward modeling of shale gas fracturing dynamic monitoring using the borehole-to-ground transient electromagnetic method

The borehole-ground transient electromagnetic method enhances the detection range and resolution by placing the transmitter electrode within the borehole, allowing for close-proximity excitation of subsurface targets and compensating for the shortcomings of traditional ground-based methods. Utilizing an unstructured vector finite element method combined with a second-order backward Euler variable time-stepping difference scheme and the MUMPS solver, we have achieved three-dimensional forward modeling simulation of the transient electromagnetic field for a borehole-ground electric source. Based on the validation of its accuracy, we analyzed the effectiveness of this method for dynamic monitoring of shale gas reservoir fracturing. Through forward modeling, we examined the characteristics of the radial electric field response during shale gas reservoir fracturing monitoring and evaluated the effectiveness of the borehole-ground TEM method in dynamic monitoring of shale gas reservoirs. The research results indicate that this method can meet the requirements for reservoir fracturing dynamic monitoring and has a promising application prospect.

Time dependent magnetic field effects on the MHD flow and heat transfer in a rectangular duct

AbstractWe consider the effect of time dependent magnetic field on the natural convection magnetohydrodynamic flow in a rectangular duct. The flow is two‐dimensional, unsteady, laminar and the fluid is electrically conducting and incompressible. The heat transfer is taken into account using the Boussinesq approximation and the buoyancy force is included in the momentum equations for the thermal coupling. The flow is considered at low magnetic Reynolds number setting and hence, the induced magnetic field is neglected. The proposed model in which the governing equations are given in terms of stream function, vorticity and temperature, is approximated by using a Chebyshev spectral collocation method for the spatial discretisation coupled with a backward difference scheme that is unconditionally stable for the temporal integration. The flow and heat transfer characteristic are analysed for several definitions of applied time varying magnetic field. Increase of the Hartmann number and the time variation of the applied magnetic field, changes the flow behaviour, especially at transient time levels and at the steady‐state.

Time-dependent simplified [formula omitted] approximation for neutron transport equation

Time-dependent transport is still too computationally expensive for various practical purposes. On the other hand, the diffusion approximation is unable to provide accurate results for many real problems. To establish a middle ground between the accuracy of the transport and the simplicity of the diffusion approximations, the simplified spherical harmonics (SPN) method was introduced. The present paper aims to improve the accuracy of the time-dependent SP3 method by employing the simplified double-P1 (SDP1) approximation, which is based on the one-dimensional double-spherical harmonics DP1 equations. Using the finite element method to treat the spatial dependence of the SDP1 equations leads to a set of first-order ordinary differential equations on the time variable. These equations are then converted into a set of linear algebraic equations by discretizing the time variable with the implicit backward finite difference scheme. Our numerical results indicate that the SDP1 approximation could compute more precise values than the widely used SP3 method in transient problems.

MLS-based numerical manifold method based on IPIM for 3D transient heat conduction of FGMs

In this study, the moving least squares based numerical manifold method (MLS-NMM) is presented to solve three-dimensional (3D) transient heat conduction problems of functionally graded materials (FGMs), where the increment dimensional precise integration method (IPIM) is applied to integrate the ordinary differential equations derived from the semi-discrete form. In the 3D MLS-NMM, the influence domains of the MLS-nodes are taken as the mathematical patches which are used to construct the mathematical cover (MC), and the shape functions of the MLS-nodes are employed as the weight functions subordinate to the MC. MLS-NMM is utilized for spatial discretization of the weak form of the problem. The IPIM overcomes the shortcomings of traditional backward difference scheme dependent on the time step and greatly improves the computing efficiency. Meanwhile, a mass lumping technique is proposed for the 3D MLS-NMM. Furthermore, a number of numerical experiments on transient thermal conduction are conducted, suggesting that the 3D MLS-NMM based on IPIM and the proposed mass lumping has excellent accuracy, absolutely stable and high computing efficiency.

Stabilized mixed finite element method for a quasistatic Maxwell viscoelastic model

This paper considers a stabilized mixed finite element method (MFE) for a quasistatic Maxwell viscoelastic model based on the L2(Ω)×H1(Ω) variational framework. The spatial discretization uses arbitrary degree piecewise polynomials Pl−Pk(l≥0,k≥1) to approximate stress and velocity. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We give a unified analysis to show the stability of the semi-discrete and fully discrete solutions and derive the corresponding priori error estimates. Numerical examples are provided to support the theoretical analysis.

Second order backward difference scheme combined with finite element method for a 2D Sobolev equation with Burgers' type non-linearity

Second order backward difference scheme combined with finite element method for a 2D Sobolev equation with Burgers' type non-linearity

Transient heat conduction modeling in continuous and discontinuous anisotropic materials with the numerical manifold method

Transient heat conduction is a common problem in engineering. However, related studies in anisotropic materials remain relatively scarce for their complexity, compared with isotropic ones. Benefiting from the use of the two cover systems, i.e., the mathematical cover and the physical cover, the numerical manifold method (NMM) provides a unified framework for both continuous and discontinuous analyses. Herein, the NMM is applied to solve transient heat conduction problems with anisotropic materials for continuous and discontinuous cases. Firstly, the fundamental formulas for anisotropic heat conduction are displayed. Then, the NMM approximations for heat conduction in continuous and discontinuous situations are given, and the discrete formulations are derived based on the Galerkin-form weighted residual method and solved by the backward difference scheme. Finally, typical numerical examples are conducted with non-conforming mathematical covers, and the results show that the proposed method can tackle both continuous and discontinuous unsteady anisotropic heat conduction problems with high accuracy and convenience.

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.

A higher‐order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions

AbstractWe propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin‐boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature‐based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first‐order in time and second‐order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.

A domain decomposition method of Schwarz waveform relaxation type for singularly perturbed nonlinear parabolic problems

We introduce a domain decomposition method of discrete Schwarz waveform relaxation (DSWR) type for a singularly perturbed nonlinear parabolic problem. The method utilizes Shishkin transition parameter for a space–time decomposition of the computational domain. In each subdomain, the problem is discretized using the central differencing and backward difference schemes on a uniform mesh in space and time directions, respectively. Further, the exchange of information between the subdomains is done through the Dirichlet data that leads to optimal convergence. We analyse the convergence of the developed method and show that the method converges very fast for small perturbation parameter and provides uniformly convergent approximations to the solution of the nonlinear problem. Finally, with some numerical experiments, we illustrate our theoretical results.

On the application of higher-order Backward Difference (BDF) methods for computing turbulent flows

The Backward Differentiation Formulae (BDF) of orders one to six are implemented in an in-house three-dimensional finite-volume (FVM) compressible flow code that operates on unstructured meshes. The lower-upper symmetric Gauss-Seidel (LUSGS) implicit relaxation technique based on the splitting of convective Jacobians is used to obtain a variable-order stable implementation in the solver. The temporal order of accuracy of the implemented schemes is verified using the Method of Manufactured Solutions (MoMS). Practical engineering flow problems are simulated to investigate the operational stability of BDF methods so that these can be used as a cheaper alternative to multi-stage methods in RANS, hybrid RANS-LES, and LES implementations. Steady-state flow simulations of supersonic flow over a flat plate and high Reynolds number flow over a NACA-0012 airfoil show that algorithm with BDF schemes of orders 1-5 is stable at high CFL numbers (700-1000) and produces a converged solution. Stokes' second problem and a high-resolution delayed detached eddy simulation (DDES) of flow over a 3D circular cylinder using the BDF1-BDF6 methods demonstrate the use of these higher-order temporal schemes in unsteady laminar and turbulent flows. It is concluded that BDF methods of orders 3-5 can be practically employed to achieve higher levels of temporal accuracy in flow simulations when the level of spatial accuracy is also high (ENO/WENO schemes, spectral methods, etc.) in hybrid RANS-LES, LES or DNS. Even without higher-order spatial accuracy, the same BDF schemes can be used in Adaptive Time-stepping (ATS) methods to obtain prescribed temporal accuracy efficiently by algorithmically switching between the different schemes.

Time-Domain Boundary Element Method with Broadband Impedance Boundary Condition

Acoustic liners are an effective tool for noise reduction and are characterized by a frequency-dependent impedance value. In this paper, a time-domain boundary element method for acoustic scattering coupled with a broadband impedance boundary condition is studied for the case of no mean flow. A Burton-Miller reformulation of the time domain boundary element method for acoustic scattering is carried out with both the pressure and its surface normal derivative terms retained. A multipole impedance model is converted into the time domain and used for enforcing an impedance boundary condition over a wide range of frequencies. Discretization of the coupled system is described. In particular, the time-domain impedance boundary condition is discretized by the third-order implicit backward difference scheme. Stability of the coupling is assessed by an eigenvalue analysis. Scattering solutions that demonstrate the validity and stability of the numerical method are presented.

All-at-once multigrid approaches for one-dimensional space-fractional diffusion equations

We focus on a time-dependent one-dimensional space-fractional diffusion equation with constant diffusion coefficients. An all-at-once rephrasing of the discretized problem, obtained by considering the time as an additional dimension, yields a large block linear system and paves the way for parallelization. In particular, in case of uniform space–time meshes, the coefficient matrix shows a two-level Toeplitz structure, and such structure can be leveraged to build ad-hoc iterative solvers that aim at ensuring an overall computational cost independent of time. In this direction, we study the behavior of certain multigrid strategies with both semi- and full-coarsening that properly take into account the sources of anisotropy of the problem caused by the grid choice and the diffusion coefficients. The performances of the aforementioned multigrid methods reveal sensitive to the choice of the time discretization scheme. Many tests show that Crank–Nicolson prevents the multigrid to yield good convergence results, while second-order backward-difference scheme is shown to be unconditionally stable and that it allows good convergence under certain conditions on the grid and the diffusion coefficients. The effectiveness of our proposal is numerically confirmed in the case of variable coefficients too and a two-dimensional example is given.

Effect of time integration scheme in the numerical approximation of thermally coupled problems: From first to third order

The advantages of using high-order time integration schemes for thermally coupled flows are assessed numerically. First-, second-, and third-order backward difference schemes are evaluated. The problem is solved in a decoupled manner using a nested iterative algorithm for the Navier–Stokes and energy equations to eliminate decoupling errors. For the space discretization, a stabilized finite element formulation of the variational multiscale type is applied to enable the use of equal order interpolation between the problem unknowns and ensure stable solutions for convection-dominated cases. The integration schemes are compared by solving the flow over a confined square including mixed heat convection in two and three dimensions. Improved numerical approximation of dynamic solutions using high-order schemes is demonstrated in the Richardson number range of 0≤|Ri|≤10 up to a Reynolds number of Re=225.

Pointwise error estimates for a system of two singularly perturbed time‐dependent semilinear reaction–diffusion equations

We present a finite difference method for a system of two singularly perturbed initial‐boundary value semilinear reaction–diffusion equations. The highest order derivatives are multiplied by small perturbation parameters of different magnitudes. The problem is discretized using a central difference scheme in space and backward difference scheme in time on a Shishkin mesh. The convergence analysis has been given, and it has been established that the method enjoys almost second‐order parameter‐uniform convergence in space and first‐order in time. Numerical experiments are conducted to demonstrate the efficiency of the method.

Thermally developing forced convection flow through porous concentric pipes annular duct

This article shows the thermally developing flow through concentric pipes annular sector duct by describing the Darcy Brinkman flow field. The cross sectional convection-diffusion terms are transformed in power law discretized form by integrating over the differential volume, whereas backward difference scheme is used in the axial direction of heat flow. With the help of semi implicit method for pressure linked equations-revised ( SIMPLE-R), we get the solution of the governing problem. The graphs of velocity profiles against R and average Nusselt number against axial distance are plotted for different values of Darcy number and geometrical configuration parameters. It has been pointed out that velocity and thermal entrance length decrease, when we decrease the value of Darcy number. By decreasing the cross section of the concentric pipes annular sector duct in the transverse direction, thermally fully developed flow region develops earlier.

Simultaneously iterative procedure based on block Newton method for elastoplastic problems

AbstractIn this article, the authors formulate elastoplastic problems as a coupled problem of the equilibrium equation and the yield condition at each material point, and develop a numerical procedure based on the block Newton method to solve the overall structure using the finite element discretization. For the integration of stress, the backward difference scheme is employed. In the conventional return mapping algorithm, the algorithmic tangent moduli are derived analytically so that it is consistent with local iterative calculation to determine internal variables. On the other hand, in the proposed block Newton method, the tangent moduli can be obtained algebraically by eliminating the internal variables, and the internal variables are also updated algebraically without local iterative calculation. The residual of the yield condition is incorporated into the linearized equilibrium equation. The proposed approach enables the errors of the equilibrium equation and the yield condition to decrease simultaneously. Some numerical examples show the validity and effectiveness of the proposed approach by comparing the results of the proposed approach with those of the conventional return mapping algorithm.

Solution of two-dimensional diffusion–advection problems for non-isotropic media with spatially variable velocity field by the boundary element method

This work presents a boundary element method formulation for the solution of the diffusion–advection problem. The formulation, developed for two-dimensional problems, for non-isotropic media, considers a spatially variable velocity field. The only way to deal with such a kind of problem is employing a steady-state fundamental solution. Consequently, the basic BEM equation presents one domain integral related to the velocity components, and another one related to the time derivative of the variable of interest, which is approximated using a backward finite difference scheme. BEM results are compared with an available analytical solution in the first part of the first example, and with the results provided by a finite element method formulation, taken as reference ones, in the second part of the first example and in the second example. From the comparisons, one can observe a good agreement between the results furnished by the proposed formulation and the analytical and reference solutions.