Crank-Nicolson Finite Difference Method Research Articles

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.

Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

Objectives: This paper aims to address the limitations of the Crank-Nicolson Finite Difference method and propose an improved version called the modified Crank-Nicolson method. Methods: Utilized implicit discretization in time and space, with parameters k = 0.001, h = 0.1, and γ = 0.1. Conducted extensive testing on various partial differential equations. Findings: Results, displayed in Table 1, showcase the method's stability and accuracy. Comparative analysis in Table 2 demonstrates the Modified Crank-Nicolson method consistently outperforming the traditional approach, reaffirming its superiority in accuracy. Novelty: The modified Crank-Nicolson method offers a significant enhancement to the traditional Crank-Nicolson finite difference method, making it a valuable tool for effectively solving partial differential equations. Keywords: Crank­Nicolson Method, Modified Crank­Nicolson Method, Finite Difference, Partial Differential Equations, Parabolic Equations, Python Software

Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

  The fractional differential equation provides a powerful mathematical framework for modeling and understanding a wide range of complex, non-local and memory-dependent phenomena in various scientific, engineering and real-world applications. Liver metastasis is a secondary cancerous tumor that develops in the liver due to the spread of cancer cells from a primary cancer originating in another part of the human body. The primary focus of this study is to understand tumor growth in the human liver, both in the presence and absence of medication therapy. To achieve this, a temporal fractional-order parabolic partial differential equation is utilized, and its analysis is carried out using numerical methods. The Caputo derivative is employed to explore the impact of medication therapy on tumor growth. To numerically solve the mathematical model, the Crank-Nicolson finite difference method has been developed. This method is selected for its remarkable attributes, including unconditional stability and second-order accuracy in both spatial and temporal dimensions. The outcomes and insights derived from this study are effectively communicated through various graphical representations. These visual aids serve as invaluable tools for comprehending the profound impact of medication therapy on the growth of liver tumors. Through the medium of these graphical depictions, one can glean a clearer and more intuitive understanding of the complex dynamics at play. The numeric solution to this intricate problem is achieved through the implementation of algorithms meticulously crafted using versatile and powerful Python programming language. Python’s flexibility, extensive libraries, and robust numerical computing capabilities make it a good choice for handling the complexities of this study.

Turbulent Lorentz heat flow visualization in radiative boundary layer regime

Modern nuclear energy systems often employ the magneto-hydrodynamic and feature radiative heat transfer. Motivated by studying the near-wall transport phenomena in such applications, the present article examines the simultaneous influence of thermal radiative flux and magneto-hydrodynamics on two-dimensional electrically conducting natural convective turbulent heat transfer about a vertical surface under a transverse static magnetic field using the low Reynolds number kinetic energy and dissipation (k – ε) model. An optimized Crank–Nicolson finite difference method is applied to solve the nonlinear and coupled system of Reynolds-averaged boundary layer equations. Numerical computations are conducted to visualize the streamlines and heatlines in laminar and turbulent regimes via mathematical stream and heat functions based on the Bejan approach. A detailed parametric study of the effects of the magnetic field, radiative flux, and turbulent Reynolds number on flow average velocity, temperature, kinetic energy, and dissipation rate is conducted. The simulations reveal that an increase in the magnetic field intensity reduces the average velocity and dissipation rate, whereas an increase in thermal radiation decreases the time mean temperature. The study also includes contour plots of turbulence energy and dissipation rate, along with skin friction coefficient and Nusselt number. The obtained numerical outcomes are compared to previous literature, and a good agreement is found.

Reconstruction of Timewise Dependent Coefficient and Free Boundary in Nonlocal Diffusion Equation with Stefan and Heat Flux as Overdetermination Conditions

     The problem of reconstruction of a timewise dependent coefficient and free boundary at once in a nonlocal diffusion equation under Stefan and heat Flux as nonlocal overdetermination conditions have been considered. A Crank–Nicolson finite difference method (FDM) combined with the trapezoidal rule quadrature is used for the direct problem. While the inverse problem is reformulated as a nonlinear regularized least-square optimization problem with simple bound and solved efficiently by MATLAB subroutine lsqnonlin from the optimization toolbox. Since the problem under investigation is generally ill-posed, a small error in the input data leads to a huge error in the output, then Tikhonov’s regularization technique is applied to obtain regularized stable results.

A Numerical Approximation on Black-Scholes Equation of Option Pricing

This paper considered the notion of European option which is geared towards solving analytical and numerical solutions. In particular, we examined the Black-Scholes closed form solution and modified Black-Scholes (MBS) partial differential equation using Crank-Nicolson finite difference method. These partial differential equations were approximated to obtain Call and Put option prices. The explicit price of both options is found accordingly. The numerical solutions were compared to the closed form prices of Black-Scholes formula. More so, comparisons of other parameters were discussed for the purpose of investment plans. The computational results shows: increase in stock volatility increases the value of options, when the initial stock price is equal to its strike price the values of call option is higher than the put option. This informs the investor about the behavior of stock prices for the purpose of decision making.  Finally, all simulation results presented graphically using MATLAB.

American strangle options with arbitrary strikes

AbstractThe so‐called American strangle options are examined in this paper. Their main characteristic is the combined put and call feature. The holder has the right to exercise prematurely choosing the option's style—put or call. We abandon the traditional assumption that the put strike is below the call one considering arbitrary values. We also assume that the put and call weights are different. The equations for the early exercise boundaries are derived in the perpetual case. After that we approximate numerically these boundaries for the finite maturity options maximizing the option holder's utility. On the basis of them we apply a Crank–Nicolson finite difference method to the corresponding Black–Scholes‐style partial differential equation to obtain the fair option price.

Coupled Effect of Chloride Corrosion and Repeated Uniaxial Compressive Loading on Unsaturated Concrete.

Concrete structure performance continuously deteriorates during operation, and the performance is simultaneously affected by chloride corrosion and repeated traffic loading. Repeated-loading-induced cracks have an impact on the rate of chloride corrosion. Chloride-induced concrete corrosion also affect the stress level of the structure under loading. Therefore, the coupled effect of repeated loading and chloride corrosion on the structural performance needs to be investigated. An upgraded test device was developed for chloride corrosion testing of unsaturated concrete structures under repeated loading. Based on the experimental results, considering the influence of repeated loading on the moisture diffusion coefficient and the chloride diffusion coefficient, a chloride transport model for unsaturated concrete under the coupled effect of repeated uniaxial compressive loading and corrosion was established. The chloride concentration under coupled loading was determined by the Crank-Nicolson finite difference method and the Thomas algorithm, and then chloride transport under the coupled effect of repeated loading and corrosion was analyzed. The results showed that the stress level and the repeated loading cycles directly affect the relative volumetric water content and chloride concentration in unsaturated concrete. The effect of chloride corrosion is more severe in unsaturated concrete compared to saturated concrete.

A modified Crank-Nicolson finite difference method preserving maximum-principle for the phase-field model

In this paper, we mainly study a new Crank-Nicolson finite difference (FD) method with a large time step for solving the nonlinear phase-field model with a small parameter disturbance. To this end, we first introduce an artificial stability term to build a modified Crank-Nicolson FD (MCNFD) scheme, and then prove that the MCNFD scheme satisfies the intrinsic maximum principle of the phase-field model. Secondly, we discuss that the MCNFD scheme satisfies the intrinsic energy stability of the phase-field model, and obtain the optimal error estimation of the MCNFD solution based on the conclusion of the discrete maximum principle. Finally, two numerical examples of one-dimensional and two-dimensional phase-field models are given to verify that the solution of the MCNFD scheme satisfies the above theoretical results.

Numerical analysis of spatial moment for colloid-facilitated contaminant transport through porous media

ABSTRACT Transport of contaminants in the presence of colloids is a three-phase problem in the groundwater porous media and colloids can also enhance the transport of contaminants by reducing the retardation factor. In this study, the numerical solution of the equilibrium model is obtained using a fully implicit Crank-Nicolson finite-difference method and validated with ten experimental results. Sensitivity and spatial moment analysis have been performed to investigate the mobility and spreading behavior of contaminants in the presence of colloids. Results suggest that enhancement of contaminant transport in the presence of colloids depends on the colloid concentration and the physical and chemical interaction of contaminants with the stationary solid matrix and suspended colloids. As the colloidal concentration increases, the mobility of contaminants augments, resulting in the early arrival of contaminants at the column outlet. The affinity of contaminants toward the stationary solid matrix of porous media can lead to the reduced transport of contaminants even under the presence of a significant amount of colloid concentration. In contrast, enhanced transport of contaminants is observed if the affinity of contaminants is more toward the suspended colloids.

Simulation of a Hybrid Thermoelectric-Magnetocaloric Refrigerator with a Magnetocaloric Material Having a First-Order Transition

A simple hybrid thermoelectric-magnetocaloric (TE-MC) system is analytically and numerically simulated using the working parameters of commercial Peltier cells and the properties of a material with a first-order and low-hysteresis magneto-structural phase transition as La(Fe,Mn,Si)13H1.65. The need for a new master equation of the heat diffusion is introduced to deal with these materials. The equation is solved by the Crank–Nicolson finite difference method. The results are compared with those corresponding to a pure TE system and a pure MC system with ideal thermal diodes. The MC material acts as a heat “elevator” to adapt its temperature to the cold or hot source making the TE system very efficient. The efficiency of the realistic hybrid system is improved by at least 30% over the pure Peltier system for the same current supply and is similar to the pure MC with ideal diodes for the same cooling power.

Introduction of the Adding and Doubling Method for Solving Bateman Equations for Nuclear Fuel Depletion

This paper introduces and evaluates the Adding and Doubling Method (ADM) for solving the Bateman equations for depletion systems with varying numbers of nuclides and compares it to the Chebyshev Rational Approximation Method (CRAM), both implemented in the reactor physics analysis application Griffin. ADM, when applied to the Crank-Nicolson Finite Difference method, can produce results comparable in accuracy and precision to CRAM with comparable run times for systems with 35 or 297 nuclides. For systems with more than 300 nuclides, the matrix-matrix operations required by ADM are significantly more costly than the matrix-vector operations required by CRAM, making CRAM the more efficient method for systems with large numbers of nuclides. ADM is an accurate method that maintains other advantages over CRAM in that it does not depend on pre-generated coefficients or require complex number operations. ADM also manages to outperform CRAM by a factor of more than 250 in terms of run time for depletion systems that require multiple Bateman solves while the depletion matrix and time step size remain constant over all depletion intervals.

A second-order accurate Crank–Nicolson finite difference method on uniform meshes for nonlinear partial integro-differential equations with weakly singular kernels

A second-order accurate Crank–Nicolson finite difference method on uniform meshes is proposed and analyzed for a class of nonlinear partial integro-differential equations with weakly singular kernels. The first-order time derivative is approximated by using a Crank–Nicolson time-stepping technique, and the singular integral term is treated by a product averaged integration rule, which preserves the positive semi-definite property of the singular integral operator. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative, and a suitable second-order discretization is adopted for the nonlinear convection term. The solvability, stability and convergence of the method are rigorously proved by the discrete energy method, the positive semi-definite property of the associated quadratic form of the method and a perturbation technique. The error estimation shows that the method has the optimal second-order convergence in time and space for non-smooth solutions. Newton’s iterative method and its algorithm implementation are presented to solve the resulting nonlinear system. Numerical results confirm the theoretical convergence result and show the effectiveness of the method.

Entropy analysis for MHD blood flow of hybrid nanoparticles (Au–Al 2 O 3 /blood) of different shapes through an irregular stenosed permeable walled artery under periodic body acceleration: Hemodynamical applications

The current study investigates the entropy generation for hybrid nanoparticles (Au–Al2O3/blood) through a vertical irregular stenosed artery in the presence of an external magnetic field, Joule heating, viscous dissipation, and heat source considering two-dimensional pulsatile blood flow and periodic body acceleration. The blood flow is assumed to be unsteady, laminar, viscous, and incompressible, and the artery walls are considered as permeable. The Reynolds temperature-dependent viscosity model is used to determine the variable viscosity effects. The governing momentum and energy equations are solved using Crank–Nicolson finite difference method by employing an appropriate coordinate transformation to build an accurate mesh using rectangular mesh units. Outcomes of the work are represented graphically for velocity, wall shear stress (WSS), volumetric flow rate, resistance impedance, temperature, heat transfer coefficient, entropy generation, and Bejan number, respectively. Also, the results are validated for velocity and temperature profiles for a given set of values of the dimensionless parameters. The entropy generation increases with rising values of shape parameter (n) up to a specific radius ( x 1 ∗ = 0.84 $x_1^*=0.84$ ) and then changes its behavior in the vicinity of the stenotic zone, in which entropy generation decreases with increasing values of shape parameter (n). The current findings may be helpful for biomedical scientists those are interested to investigate the treatment of various cardiovascular diseases (CVDs).

Effects of heat transfer, body acceleration and hybrid nanoparticles (Au–Al2O3) on MHD blood flow through a curved artery with stenosis and aneurysm using hematocrit-dependent viscosity

The present study investigates the heat transfer and body acceleration effects on unsteady MHD blood flow through a curved artery in the presence of stenosis and aneurysm using hybrid nanoparticles (). For realistic behavior of blood flow, blood is assumed to be a pulsatile, laminar, incompressible, viscous fluid with hematocrit-dependent viscosity. A uniform magnetic field is applied in the radial direction to the blood flow with radiation effect. The governing momentum and energy equations are solved using Crank-Nicolson Finite difference method through an accurate mesh using rectangular mesh units. The obtained data for both stenotic and aneurysm segments are graphically depicted and analyzed for various physical parameters. To examine the overall behavior of blood flow patterns, velocity contours for several emergent parameters have been displayed. Using clinically realistic hemodynamic data, comprehensive solutions for gold and gold- aluminium oxide hybrid mediated blood flow are presented. It is observed that the velocity profile rises as the volume fraction of gold () nanoparticles increases, while, the velocity profile decreases as the volume fraction of aluminium oxide () nanoparticles increases. It is noted that impedance of the flow decreases with increment in body acceleration parameter , while, WSS increases with increment in . The resistive impedance is higher for curved arteries than straight artery. The findings show that hybrid nanoparticles () reduce the values of hemodynamic parameters such as wall shear stress and resistive impedance, and blood flow rate decreases with an increment in and magnetic field strength. The current findings are consistent with recent findings in other studies. The outcomes of the study may be helpful in the diagnosis and treatment of plaque rupture, cancer, infections, brain aneurysms, and blood clot removal. Results may also be used to design and analyze biomedical devices for great potential treatment modalities, the anticancer drug industry, and nano-drug delivery systems.

Neural Schrödinger Equation: Physical Law as Deep Neural Network.

We show a new family of neural networks based on the Schrödinger equation (SE-NET). In this analogy, the trainable weights of the neural networks correspond to the physical quantities of the Schrödinger equation. These physical quantities can be trained using the complex-valued adjoint method. Since the propagation of the SE-NET can be described by the evolution of physical systems, its outputs can be computed by using a physical solver. The trained network is transferable to actual optical systems. As a demonstration, we implemented the SE-NET with the Crank-Nicolson finite difference method on Pytorch. From the results of numerical simulations, we found that the performance of the SE-NET becomes better when the SE-NET becomes wider and deeper. However, the training of the SE-NET was unstable due to gradient explosions when SE-NET becomes deeper. Therefore, we also introduced phase-only training, which only updates the phase of the potential field (refractive index) in the Schrödinger equation. This enables stable training even for the deep SE-NET model because the unitarity of the system is kept under the training. In addition, the SE-NET enables a joint optimization of physical structures and digital neural networks. As a demonstration, we performed a numerical demonstration of end-to-end machine learning (ML) with an optical frontend toward a compact spectrometer. Our results extend the application field of ML to hybrid physical-digital optimizations.

A numerical comparative analysis between crank-Nicolson finite difference method and binomial model for European call option price

There are several kinds of numerical techniques for solving option valuation problems. In this article Binomial model (BM) and Crank-Nicolson finite difference (CNFD) approach are applied and compared with the Black-Scholes analytic solution (BSAS) to determine the best numerical method. It has been noticed that, for the valuation of European call (EC) options at various factors, the Binomial model (BM) is found to be more accurate than the Crank-Nicolson finite difference (CNFD) technique. In addition, for the impact of high volatility, the technique of Crank-Nicolson finite difference (CNFD) approaches quicker to Black-Scholes analytic solution (BSAS) than the Binomial model (BM). Furthermore, in comparison to the Binomial model (BM), CNFD method consumes more time to determine the option price in each time step.

Retrieval of Timewise Coefficients in the Heat Equation from Nonlocal Overdetermination Conditions

     This paper investigates the simultaneous recovery for two time-dependent coefficients for heat equation under Neumann boundary condition. This problem is considered under extra conditions of nonlocal type. The main issue with this problem is the solution unstable to small contamination of noise in the input data. The Crank-Nicolson finite difference method is utilized to solve the direct problem whilst the inverse problem is viewed as nonlinear optimization problem. The later problem is solved numerically using optimization toolbox from MATLAB. We found that the numerical results are accurate and stable.

A novel regularized model for the logarithmic Klein-Gordon equation

In this paper, the energy regularization technique proposed in [Bao et al., arXiv:2006.05114] is adopted for the logarithmic Klein-Gordon equation to avoid the singularity of the logarithmic function at the origin. The convergence rate between the energy functions of the energy regularized logarithmic Klein-Gordon equation (ERLogKGE) and the logarithmic Klein-Gordon equation (LogKGE) is proved to be O(ε2). An energy preserving Crank-Nicolson finite difference method and an explicit finite difference method are presented for the ERLogKGE. By adopting the mathematical induction, energy method, and inverse inequality, the error bound of the explicit finite difference method is estimated in terms of mesh size h, time step τ, and the regularization parameter ε. For comparison reasons, two regularization techniques for the logarithmic nonlinearity are also studied for the LogKGE. A collection of numerical experiments show that the proposed energy regularization technique is superior to other regularized schemes.

Solvability of the Nonlocal Inverse Parabolic Problem and Numerical Results

In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. In addition, the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both analytical and numerically simulated noisy input data are inverted. The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the choice of the regularization parameter is also discussed based on the trial and error technique.