Crank-Nicolson Research Articles

An efficient data assimilation algorithm using the Allen–Cahn equation

Data assimilation combines observation data and the physical model to provide an estimate of the state which is more accurate than what could be obtained by using just the data or the model alone. We will establish a robust algorithm for the Allen–Cahn (AC) equation by considering the data assimilation term. The proposed approach includes a feedback control scheme for the AC equation, which forces the numerical solutions to preserve the observed data. We utilize the Crank–Nicolson formula and a stable numerical method to solve the data assimilation method using the AC equation. The proposed numerical algorithm has second-order temporal and spatial accuracy. Our computational scheme has a low computational burden because of the linearity of the elliptic equation. The discrete system of equations is unconditionally energy stable. The efficiency of our proposed method has been verified through a number of computational tests. The proposed approach is the first simple algorithm to consider the data assimilation method using the AC equation. The computational results indicate that our method performs well with low spatial resolution observational measurements and can be used for studying the image inpainting and shape transformation processing.

Theoretical analysis of unsteady buoyant turbulent heat and mass transport from a vertical plate using LRN k-ϵ model

The contemporary analysis has been developed to investigate the transient natural convective combined heat and mass transport of a turbulent flow adjacent to the plate by utilizing the low Reynolds number (LRN) k-ϵ model. The research problem on turbulent flow is assumed to be two dimensional and viscous incompressible. The governing turbulent equations such as continuity, momentum, energy, concentration, turbulent kinetic energy (TKE) and dissipation rate of TKE are considered as per the flow geometry. Due to the non-availability of analytical or direct numerical techniques, the corresponding highly nonlinear PDE’s are solved by adopting the standard implicit type of finite difference method namely Crank–Nicolson scheme. Also, this scheme is unconditionally stable and helps to keep the governing turbulent flow equations in discretized form and are solved via tridiagonal matrix algorithm. The simulated results for several control parameters including turbulent Reynolds number are graphically displayed and also analyzed the flow profiles. To understand the physical phenomenon of turbulent flow, the authors shown the turbulent behavior of average momentum, energy and mass transfer rates. It has been noted that, the average temperature and concentration fields boosted with rising values of . Also, TKE and dissipation rate of TKE profiles increase with enhancing turbulent values. In addition, the simulated turbulent flow results from the LRN k-ϵ model are compared with usual laminar flows as a special case and found to be in respectable agreement.

Numerical study of a transient MHD flow across an oscillating vertical plate with thermal radiation and viscous dissipation

The flow of an electrically conducting fluid across a vertically positioned oscillating semi-boundless plate with uniform mass diffusion and temperature is examined in this study in terms of the effects of thermal radiation and viscous dissipation. The dimensionless governing equations were solved using an effective and unconditionally stable implicit finite-difference approach known as the Crank-Nicolson method. Based on the numerical results, the impacts of various physical parameter values on concentration, temperature; velocity; Sherwood numbers, Nusselt numbers and skin-friction profiles are displayed graphically and their consequences thoroughly analyzed. We observed that when the magnetic field, radiation and phase angle parameters are increased, the velocity is reduced. This shows that plate oscillation, radiation and magnetic fields affect the flow pattern significantly.

Solve High-Dimensional Reflected Partial Differential Equations by Neural Network Method

Reflected partial differential equations (PDEs) have important applications in financial mathematics, stochastic control, physics, and engineering. This paper aims to present a numerical method for solving high-dimensional reflected PDEs. In fact, overcoming the “dimensional curse” and approximating the reflection term are challenges. Some numerical algorithms based on neural networks developed recently fail in solving high-dimensional reflected PDEs. To solve these problems, firstly, the reflected PDEs are transformed into reflected backward stochastic differential equations (BSDEs) using the reflected Feyman–Kac formula. Secondly, the reflection term of the reflected BSDEs is approximated using the penalization method. Next, the BSDEs are discretized using a strategy that combines Euler and Crank–Nicolson schemes. Finally, a deep neural network model is employed to simulate the solution of the BSDEs. The effectiveness of the proposed method is tested by two numerical experiments, and the model shows high stability and accuracy in solving reflected PDEs of up to 100 dimensions.

A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10,11,34] for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.

Fitted cubic spline in tension difference scheme for two-parameter singularly perturbed delay parabolic partial differential equations

A numerical study of a two-parameter singularly perturbed time-delay parabolic equation has been initiated. The proposed technique is based on a fitted operator finite difference scheme. The first step involves the discretization of the time variables using the Crank–Nicolson method. This results in singularly perturbed semi-discrete problems, which are further discretized in space using the exponentially fitted tension-spline finite difference method. The error analysis is carried out. It is shown to be parameter-uniformly convergent and second-order accurate. To support the theoretical results, numerical experiments are carried out by applying the proposed technique to two test examples.

The Convergence of Symmetric Discretization Models for Nonlinear Schrödinger Equation in Dark Solitons’ Motion

The Schrödinger equation is one of the most basic equations in quantum mechanics. In this paper, we study the convergence of symmetric discretization models for the nonlinear Schrödinger equation in dark solitons’ motion and verify the theoretical results through numerical experiments. Via the second-order symmetric difference, we can obtain two popular space-symmetric discretization models of the nonlinear Schrödinger equation in dark solitons’ motion: the direct-discrete model and the Ablowitz–Ladik model. Furthermore, applying the midpoint scheme with symmetry to the space discretization models, we obtain two time–space discretization models: the Crank–Nicolson method and the new difference method. Secondly, we demonstrate that the solutions of the two space-symmetric discretization models converge to the solution of the nonlinear Schrödinger equation. Additionally, we prove that the convergence order of the two time–space discretization models is O(h2+τ2) in discrete L2-norm error estimates. Finally, we present some numerical experiments to verify the theoretical results and show that our numerical experiments agree well with the proven theoretical results.

Local discontinuous Galerkin method for the Riesz space distributed-order Sobolev equation

We present and analyze a local discontinuous Galerkin finite element method to solve the Riesz space distributed-order Sobolev equation. To approximate the distributed-order Riesz space derivative, the Gauss quadrature as the high computational accuracy method is proposed. A multi-term fractional equation is then constructed from the considered equation by approximating the Riesz space derivative. Moreover, stability analysis is provided for a semi-discrete scheme. We provide numerical results to justify the theoretical analysis by solving the ordinary differential equation. We obtain this after implementing the local discontinuous Galerkin scheme on the Riesz space distributed-order Sobolev equation with the Crank–Nicolson scheme as a time marching method and the Laplace transform technique.

An alternating direction implicit compact finite difference scheme for the multi-term time-fractional mixed diffusion and diffusion–wave equation

An alternating direction implicit(ADI) compact finite difference scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion–wave equation is given in this paper. By using the Riemann–Liouville fractional integral operator on both sides of the original equation, we obtain a time-fractional integro–differential equation with the order of the highest derivative being one. Using the weighted and shifted Grünwald formulas of Riemann–Liouville fractional derivative and fractional integral, and the Crank–Nicolson approximation, a temporal second-order approximation is obtained for the equivalent integrodifferential system. In the spatial direction, a fourth-order compact approximation is employed to give a fully discrete scheme. Using the splitting techniques for higher dimensional problems, we derive the fully discrete ADI Crank–Nicolson scheme. With the positive definiteness of the related time discrete coefficients and spatial operators, the stability and convergence (second-order in time and fourth-order in space) in the discrete L2-norm are proved by using energy method. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.

A unified approach to maximum-norm a posteriori error estimation for second-order time discretizations of parabolic equations

Abstract A class of linear parabolic equations is considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretizations combined with finite element discretizations in space. In particular, we study the Crank–Nicolson method, the extrapolated Euler method, the backward differentiation formula of order 2, the Lobatto IIIC method and a two-stage SDIRK method. We use the idea of elliptic reconstructions and certain bounds for the Green’s function of the parabolic operator.

Disturbances in system dynamics of [Formula: see text] and [Formula: see text] perturbing insulin secretion in a pancreatic [Formula: see text]-cell due to type-2 diabetes.

The individual study of [Formula: see text] and [Formula: see text] dynamics respectively in a [Formula: see text]-cell has yielded limited information about the cell functions. But the systems biology approaches for such studies have received very little attention by the research workers in the past. In the present work, a system-dynamics model for the interdependent [Formula: see text] and [Formula: see text] signaling that controls insulin secretion in a [Formula: see text]-cell has been suggested. A two-way feedback system of [Formula: see text] and [Formula: see text] has been considered and one-way feedback between [Formula: see text] and insulin has been implemented in the model. The finite element method along with the Crank-Nicolson method have been applied for simulation. Numerical results have been used to analyze the impact of perturbations in [Formula: see text] and [Formula: see text] dynamics on insulin secretion for normal and Type-2 diabetic conditions. The results reveal that Type-2 diabetes comes from abnormalities in insulin secretion caused by the perturbation in buffers and pumps (SERCA and PMCA).

An effective second-order scheme for the nonstationary incompressible magnetohydrodynamics equations

We are proposing the Crank-Nicolson/Adams-Bashforth (CN/AB) algorithm for the nonstationary magnetohydrodynamic (MHD) equations. For time discretization, the time derivative terms are approximated by a first-order Euler-backward scheme, an implicit second-order Crank-Nicolson scheme acting on linear terms, and an explicit Adams-Bashforth scheme dealing with nonlinear terms. We use the finite element method for spatial discretization. Theoretical studies show that the proposed algorithm is almost unconditionally stabilized with τ≤c. The optimal error estimate of the algorithm is obtained by a parabolic argument mathematical trick. Finally, the theoretical results are verified by numerical experiments.

Numerical solutions of free convective flow from a moving cone in the presence of non-uniform surface temperature

This computational study intends to determine the impact of non-isothermal surface temperature on the natural convective boundary layerflow of fluid along a moving vertical cone. The governing equation deals with continuity, momentum, and energy along with thermal fluctuations are reduced into dimensionless unsteady non-linear coupled PDE. The non-dimensional governing equations of the boundary layer are then illuminated by a systematic and unconditionally stable method of implicit finite difference Crank-Nicolson scheme. Analyzing the stability and convergence of the system has helped in the optimization of parameters and accuracy of the system. Using a tabular approach, we validate the present numerical methodology by comparing it to the previously discussed results. Through a table of values, the physical impacts of parameters are analyzed and graphical representations are provided. From the graphs, it is evident that an increment in Pr and n results in lower temperature and velocity profiles.

Sensitivity analysis of electroosmotic magnetohydrodynamics fluid flow through the curved stenosis artery with thrombosis by response surface optimization

The current numerical investigation focuses on the sensitivity analysis of the pertinent parameters of the electrically conducting fluid (blood) with the suspension of Al2O3 nanoparticles through the constricted arteries in the presence of stenosis, thrombosis and catheter. The mathematical model is developed to simulate the blood flow through the catheterized artery having irregular stenosis at the wall and thrombosis at the centre. The interactions of different physical phenomena like electroosmosis, radiation, Joule heating and a uniform radial magnetic field actuate the fluid flow. The nanoparticles shape effect is taken into account by considering the Crocine model. The governing equations are discretized by employing the Crank-Nicolson scheme, and the graphical portrayals for the impact of significant parameters on the flow characteristics have been devised and discussed. The statistical experiment is designed based on the parametric range of the parameter. The sensitivity analysis is performed on response surfaces after getting the best-fitted response surface relative to each response variable. The WSS profile negatively correlates with hematocrit parameter hm, Hartmann number M and positively related with Debye–Huckel parameter qe, while the reverse trend is observed for the impedance profile. The present work may be helpful in biomedical engineering, such as magnetic resonance angiography (MRA), which creates artery images. The integrated medical imaging and hemodynamic framework assist surgeons and researchers in better understanding the abnormalities caused by cardiovascular diseases and gaining a deeper grasp of the hemodynamic flow and other bodily fluids for surgical restoration.

Thermal description and entropy evaluation of magnetized hybrid nanofluid with variable viscosity via Crank–Nicolson method

The dominant characteristics of hybrid nanofluids, such as low cost, improved heat transfer rates, and higher thermal and electrical conductivity, make them preferable fluids in thermal energy systems. In light of these incredible features, our goal in the present research is to analyse heat transfer and entropy generating in (Fe3O4–Cu)/water hybridity nano liquid flowing via a vertical cone with variable wall temperature inserted in a porous material. The effects of variable viscosity, magnetic force, and thermal radiative flux are additional aspects that contribute to the originality of the constructed model. The mathematical model is solved utilising the Crank-Nicolson technique, and the numerical findings are showed graphically and in a tabular manner. The heat transfer process is improved by hybrid nanoparticles and porosity while being hindered by magnetic interaction, viscosity, and thermal radiation. Energy loss in the form of increased entropy can be noted at the further vertical end of the cone than the base. The parametrical influence was reported to be nominal compared to the other physical aspects. Domination of heat transfer induced entropy generation and be observed for the porosity improvements.

Thermal conductivity reconstruction method with application in a face milling operation

PurposeThis paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.Design/methodology/approachThe discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.FindingsThe paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.Originality/valueThe paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

A nonconventional stability approach for a nonlinear Crank–Nicolson method solving degenerate Kawarada problems

Conventionally, the numerical stability of finite difference approximations of nonlinear Kawarada problems is shown only via frozen source terms, that is, ignoring potential jeopardization from quenching nonlinearities. The approach leaves an inadequacy behind even in the sense of localized stability analysis. This paper provides a much improved analysis of the numerical stability without freezing nonlinear source terms of the underlying equations. The strategy implemented can be extended for similar endeavors to higher dimensional cases. Simulation experiments are included.

Device Performance of a Tubular Membrane Dialyzer Incorporating Ultrafiltration Effects on the Dialysis Efficiency.

Membrane dialysis is one of the membrane contactors applied to wastewater treatment. The dialysis rate of a traditional dialyzer module is restricted because the solutes transport through the membrane only by diffusion, in which the mass-transfer driving force across the membrane is the concentration gradient between the retentate and dialysate phases. A two-dimensional mathematical model of the concentric tubular dialysis-and-ultrafiltration module was developed theoretically in this study. The simulated results show that the dialysis rate improvement was significantly improved through implementing the ultrafiltration effect by introducing a trans-membrane pressure during the membrane dialysis process. The velocity profiles of the retentate and dialysate phases in the dialysis-and-ultrafiltration system were derived and expressed in terms of the stream function, which was solved numerically by the Crank-Nicolson method. A maximum dialysis rate improvement of up to twice that of the pure dialysis system (Vw=0) was obtained by employing a dialysis system with an ultrafiltration rate of Vw=2 mL/min and a constant membrane sieving coefficient of θ=1. The influences of the concentric tubular radius, ultrafiltration fluxes and membrane sieve factor on the outlet retentate concentration and mass transfer rate are also illustrated.

An efficient and robust Lagrange multiplier approach with a penalty term for phase-field models

In this paper, we propose and analyze a variant of Lagrange multiplier approach with a penalty term for phase-field models. More precisely, we develop two types of linear second-order decoupled and unconditionally energy stable time-stepping schemes for phase-field models based on the Crank-Nicolson (CN) and the second order backward differentiation formulations (BDF2), where the scalar multiplier variable is evaluated by solving a nonlinear algebraic equation. Comparing with the Lagrange multiplier method (Cheng et al. (2020) [6]) for phase-field models, a scalar penalty term is first introduced to improve the time-step size constraint for the convergence of Newton's iterative algorithm for the nonlinear algebraic equation, which allows us to obtain robust and efficient numerical schemes for phase-field models. Moreover, we only use first order time-stepping scheme to approximate the scalar multiplier variable, which will not affect the second order accuracy of the unknown phase function with some special Lagrange multiplier functions. Meanwhile, it plays an essential role in the derivation of energy stability of the proposed BDF2-based scheme on the nonuniform temporal mesh. In addition, the energy stability of the CN scheme is rigorously established. Furthermore, we deduce that the proposed BDF2 scheme is energy stable with the adjacent time-step ratios no more than 4.8645. Finally, a series of numerical tests are presented to numerically validate the efficiency and stability of our proposed Lagrange multiplier approach.

Lack of robustness and accuracy of many numerical schemes for phase-field simulations

In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small. More specifically, by rigorous analysis in most cases, we have the following conclusions: (i) The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution. (ii) The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions. (iii) For the second-order fully implicit and convex splitting schemes, for any time step size [Formula: see text], there exists an initial condition [Formula: see text], with [Formula: see text], such that the numerical solution converges to the wrong steady state solution. (iv) For [Formula: see text], all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small. (v) An unconditionally energy-stable scheme (such as the modified Crank–Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank–Nicolson scheme). Most, if not all, of the above conclusions are expected to be true for more general Allen–Cahn and other phase-field models.