Crank-Nicolson Finite Difference Method Research Articles

Supply-Side Perspective for Carbon Pricing

This paper theoretically and empirically revisits carbon pricing from the supply-side perspective for carbon assets to solve a recent low price issue, which may delay the development of emission reduction technologies in the sense of marginal abatement costs. We propose a carbon pricing model linked to crude oil prices, which has historically been employed in supply-side driven pricing of long-term contracts for energy trading in the early stage. Since the model is designed to hold carbon prices between certain lower and upper boundaries using S-shaped carbon price linkage to crude oil prices, it can be useful to overcome a recent low carbon price issue. In addition, it is shown that the model can alleviate the difficulties of carbon derivative pricing in selecting market price of risk. Empirical studies using EUA and Brent crude oil futures prices estimate the parameters of the Brent crude oil-linked EUA price model. EUA prices simulated from the model with the parameter estimates are compared with historical EUA prices. The results suggest that simulated EUA prices from the model be kept relatively higher than historical EUA prices. This is preferable for accelerating carbon emission reductions in that emission reduction technologies with high marginal abatement costs are affordable. It may imply that EUA must be priced using a crude oil-linked carbon price model in the early stage of EUA trading until EUA markets are fully matured from the supply-side perspective for carbon asset pricing, not employing a premature market-based or supply and demand-based carbon price model. To show usefulness of crude oil-linked carbon pricing, we also give a numerical example of European carbon option pricing based on the Brent crude oil-linked EUA price model by using Crank-Nicolson finite difference method. Finally we discuss the relation between crude oil-linked carbon pricing and emission reduction risk. These studies may suggest carbon policy makers take into account of crude oil-linked carbon pricing to tackle low price and low liquidity issues of carbon assets.

Heat Transfer in a Porous Radial Fin: Analysis of Numerically Obtained Solutions

A time dependent nonlinear partial differential equation modelling heat transfer in a porous radial fin is considered. The Differential Transformation Method is employed in order to account for the steady state case. These solutions are then used as a means of assessing the validity of the numerical solutions obtained via the Crank-Nicolson finite difference method. In order to engage in the stability of this scheme we conduct a stability and dynamical systems analysis. These provide us with an assessment of the impact of the nonlinear sink terms on the stability of the numerical scheme employed and on the dynamics of the solutions.

Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation

A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion equation in the case where the Grunwald-Letnikov definition is used for the time-fractional derivative. The stability and convergence of the proposed Crank-Nicolson scheme are also analyzed. Finally, numerical examples are presented to test that the numerical scheme is accurate and feasible.

Numerical Simulation of Groundwater Pollution Problems Based on Convection Diffusion Equation

The analytical solution of the convection diffusion equation is considered by two-dimensional Fourier transform and the inverse Fourier transform. To get the numerical solution, the Crank-Nicolson finite difference method is constructed, which is second-order accurate in time and space. Numerical simulation shows excellent agreement with the analytical solution. The dynamic visualization of the simulating results is realized on ArcGIS platform. This work provides a quick and intuitive decision-making basis for water resources protection, especially in dealing with water pollution emergencies.

Unsteady MHD Free Convective Flow Past an Inclined Parabolic Accelerated Plate with Hall Current, Radiation Effects and Variable Temperature in a Porous Medium

We analyze the effects of hall current and radiation on unsteady hydromagnetic free convection flow of a viscous incompressible electrically conducting fluid past an inclined parabolic accelerated plate in a porous medium by applying a transverse magnetic field which makes an angle α to the inclined plate.The fluid is assumed to be viscous, incompressible and electrically conducting with a strong magnetic field. Using the modified Ohm’s law and the Bossinesq approximation the governing equations of the problem are reduced to local non-similarity boundary layer equations using suitable transformation.The dimensionless governing equations of flow field are solved numerically by Crank-Nicolson finite difference method for different values of governing flow parameters. The velocity and temperature profiles are shown through graphs.

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.

Magnetohydrodynamic Free Convection Flow with Thermal Radiation and Chemical Reaction Effects in the Presence of Variable Suction

The aim of the present study is to investigate the effect of flow parameters on the free convection and mass transfer of an unsteady magnetohydrodynamic flow of an electrically conducting, viscous and incompressible fluid past an infinite vertical porous plate in the presence of variable suction. The thermal radiation and chemical reaction effects are assumed to exist within the channel. Non dimensional partial differential equations of governing equations of flow are solved numerically using Crank Nicolson finite difference method. The skin friction, heat and mass transfer rates as well as the effects of various parameters on velocity, temperature and concentration profiles are analyzed. The signifiant results from this study are that an increase in the values of radiation parameter and chemical reaction parameter causes a reduction in the velocity, temperature and concentration.

Effects of Thermophoresis, Dufour, Hall and Radiation on an Unsteady MHD Flow past an Inclined Plate with Viscous Dissipation, Chemical Reaction and Heat Absorption and Generation

This paper investigates study of MHD(Magnetohydrodynamics) flow of viscous incompressible fluid past an inclined porous plate embedded in porous medium with the effects of Thermophoresis, Dufour, Hall, radiation, viscous dissipation, chemical reaction and heat generation or absorption. Non dimensional partial differential equations of governing equations of flow are solved numerically by applying Crank-Nicolson finite difference method for different values of parameters. Velocity, temperature, concentration profiles are discussed through graphs for different values of parameters and skin friction coefficients, Nusselt number and Sherwood number are discussed through tables.

Iterative solvers for image denoising with diffusion models: A comparative study

In this paper we propose and compare the use of two iterative solvers using the Crank–Nicolson finite difference method, to address the task of image denoising via partial differential equations (PDEs) models such as Regularized Perona–Malik equation or C-model and Bazan model (Bilateral-filter-based model). The solvers which are considered in this paper are the Successive-over-Relaxation (SOR) and an advanced solver known as Hybrid Bi-Conjugate Gradient Stabilized (Hybrid BiCGStab) method. From numerical experiments, it is found that the Crank–Nicolson method with hybrid BiCGStab iterative solver produces better results and is more efficient than SOR and already existing, in terms of MSSIM and PSNR.

Investigations on several numerical methods for the non-local Allen–Cahn equation

This paper investigates some numerical methods for solving the non-local Allen–Cahn equation with a space–time dependent Lagrange multiplier. Several types of methods, including the Crank–Nicolson finite difference method, the finite difference operator splitting method, and the Fourier spectral operator splitting method are proposed respectively. Comparisons are made among these methods in terms of accuracy and computational efficiency for solving different types of problems. Numerical results show that the Fourier spectral operator splitting method is very efficient because of its spectral accuracy in space, especially for simulation of long time dynamics.

Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation

The aim of the present paper is to present a numerical algorithm for the time-dependent generalized regularized long wave equation with boundary conditions. We semi-discretize the continuous problem by means of the Crank–Nicolson finite difference method in the temporal direction and exponential B-spline collocation method in the spatial direction. The method is shown to be unconditionally stable. It is shown that the method is convergent with an order of . Our scheme leads to a tri-diagonal nonlinear system. This new method has lower computational cost in comparison to the Sinc-collocation method. Finally, numerical examples demonstrate the stability and accuracy of this method.

Numerical modeling considerations for an applied nonlinear Schrödinger equation.

A model for nonlinear optical propagation is cast into a split-step numerical framework via a variable stencil-size Crank-Nicolson finite-difference method for the linear step and a choice of two different nonlinear integration schemes for the nonlinear step. The model includes Kerr, Raman scattering, and ionization effects (as well as linear and nonlinear shock, diffraction, and dispersion). We demonstrate the practical importance of numerical effects when interpreting computational studies of high-intensity optical pulse propagation in physical materials. Examples demonstrate the significant error that can arise in discrete, limited precision implementations as one attempts to improve practical operator accuracy through increased operator support size and sampling frequency. We also demonstrate the effect of the method used to obtain the finite-difference operator coefficients defining the equations ultimately used in the discrete model. Smooth, plausible, but incorrect solutions may result from these numerical effects. This implies the necessity of a complete, precise description of all numerical methods when reporting results of computational physics investigations in order to ensure proper interpretation and reproducibility.

MELLIN TRANSFORM METHOD FOR THE VALUATION OF SOME VANILLA POWER OPTIONS WITH NON-DIVIDEND YIELD

This paper presents the Mellin transform method for the val- uation of some vanilla power options with non-dividend yield. This method is a powerful tool used in the valuation of options. We extend the Mellin transform method proposed by Panini R. and Srivastav R.P. (15) to derive the price of European and American power put options with non-dividend yield. We also derive the fundamental valuation formula known as the Black-Scholes model using the convolution property of the Mellin transform method. To pro- vide a sufficient numerical analysis, we compare the results generated by the Mellin transform method for the valuation of American power put option for n = 1 which pays no dividend yield to two other numerical methods namely Crank Nicolson finite difference method (2) and binomial model (3) for options valuation against Black-Scholes analytical pricing formula (1). The numerical experiment shows that the Mellin transform method is efficient, easy to imple- ment, agree with the values of Black-Scholes (1), Crank Nicolson finite difference method (2) and binomial model (3). Hence the Mellin transform method is a better alternative method compared to the Crank Nicolsion finite difference and binomial model for the valuation of some vanilla power options.

A parallel Crank–Nicolson finite difference method for time-fractional parabolic equation

Abstract- In this paper, a parallel Crank-Nicolson finite difference method (C-N-FDM) for time-fractional parabolic equation on a distributed system using MPI is investigated. The fractional derivative is described in the Caputos sense. The resultant large system of equations is studied using preconditioned conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiment using different number of nodes of the Linux PC cluster is studied. The performance metrics clearly show the benefit of using the proposed approach on the Linux PC cluster in terms of execution time reduction and speedup with respect to the sequential running in a single PC.

Numerical Methods and Simulations for the Dynamics of One-Dimensional Zakharov–Rubenchik Equations

In this paper, we propose and study several accurate numerical methods for solving the one-dimensional Zakharov–Rubenchik equations (ZRE). We begin with a review on the important properties of the ZRE, including the solitary wave solutions and the various conservation laws. Then we propose a very efficient and accurate numerical method based on the time-splitting technique and the Fourier pseudo-spectral (TSFP) method. Next, we propose some conservative and non-conservative types of finite difference time domain methods, including a Crank–Nicolson finite difference method that conserves the mass and the energy of the system in the discrete level. Discrete conservation laws and numerical stability of all the proposed methods are analyzed. Comparisons between different methods in the efficiency, stability and accuracy are carried out, which identifies that the TSFP method is the most efficient and accurate numerical method among all the methods. Lastly, we apply the TSFP method to simulate and study the dynamics of the solitons in the ZRE numerically.

Numerical investigations on self-similar solutions of the nonlinear diffusion equation

In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation ht=−(h3hxxx)x, which arises in the context of surface-tension-driven flow of a thin viscous liquid film. Here, h=h(x,t) is the liquid film height. A self-similar solution is h(x,t)=h(α(t)(x−x0)+x0,t0)=f(α(t)(x−x0)) and α(t)=[1−4A(t−t0)]−1/4, where A and x0 are constants and t0 is a reference time. To discretize the governing equation, we use the Crank–Nicolson finite difference method, which is second-order accurate in time and space. The resulting discrete system of equations is solved by a nonlinear multigrid method. We also present efficient and accurate numerical algorithms for calculating the constants, A, x0, and t0. To find a self-similar solution for the equation, we numerically solve the partial differential equation with a simple step-function-like initial condition until the solution reaches the reference time t0. Then, we take h(x,t0) as the self-similar solution f(x). Various numerical experiments are performed to show that f(x) is indeed a self-similar solution.

A Crank–Nicolson Galerkin approach to the analysis of electromechanical oscillations in stressed power systems

The primary goal of this research is directed toward the study and characterization of nonlinear power system transient phenomena following large perturbations. A novel analytical framework based on a nonlinear Galerkin method and a finite difference scheme is introduced. The proposed method is designed primarily to capture strong nonlinear behavior and may be applied to model reduction, nonlinear characterization and control design.A nonlinear system model is first developed by Galerkin projections onto basis of eigenfunctions obtained from the proper orthogonal decomposition (POD) method that includes the representation of higher-order nonlinear effects. The scheme is coupled with a modified Crank–Nicolson finite difference method to characterize the nonlinear system dynamics following severe perturbations. Several extensions and generalizations of the method are suggested and tested and criteria are then developed for systematically analyzing complicated system oscillations.The theory is illustrated on two test power systems. Results obtained using the Galerkin procedures are compared with transient stability simulations. Factors that influence the quality of the nonlinear Galerkin approximations are discussed and issues concerning the implementation of the method and numerical calculations are also highlighted.

Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

We establish uniform error estimates of finite difference methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter $\varepsilon$ ($\varepsilon\in(0,1]$). When $\varepsilon\to0^+$, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., $0<\varepsilon\ll1$, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with $O(\varepsilon^2)$-wavelength at $O(\varepsilon^4)$ and $O(\varepsilon^2)$ amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in $\varepsilon$ of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in $\varepsilon$, at the order of $O(h^2+\tau)$ and $O(h^2+\tau^{2/3})$ with time step $\tau$ and mesh size $h$ for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the $l^2$-norm and discrete semi-$H^1$ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine $\varepsilon$-dependent error estimates of NLSW, $\varepsilon$-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-$H^1$ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.

An almost second order fitted mesh numerical method for a singularly perturbed delay parabolic partial differential equation

In this paper we develop a numerical method for solving a singularly perturbed delay parabolic partial differential equation. The proposed method consists of Crank-Nicolson finite difference method constructed on a mesh of Shishkin type and hence referred to as a fitted mesh finite difference method. We analyzed the method for stability and convergence and found that it is unconditionally stable and converges with order O (Nt-2 + Nx-2ln2Nx) where Nt and Nx are the numbers of subintervals in the t and x directions, respectively. The performance of the method is illustrated through numerical experiments.

Mathematical Modeling and Simulation of Microwave Thawing of Large Solid Foods Under Different Operating Conditions

Microwaves require shorter times to increase foodstuffs temperature when compared to conventional heating methods. However, there are some problems associated to temperature distribution within the products, owing to the preferential absorption of electromagnetic energy by liquid water, caused by differences between its dielectric properties and those of ice (“runaway”). To analyze the behavior of food microwave thawing, a mathematical three-dimensional (3D) model was developed by solving the unsteady-state heat and mass transfer differential equations; this model can be applied to large systems for which Lambert’s law is valid. Thermal, mass transport, and electromagnetic properties varying with temperature were used. The numerical solution was developed using an implicit Crank–Nicolson finite difference method using the classical formulation for one-dimensional (1D) systems and the alternating direction method in two and three dimensions. The model was validated using experimental data from the literature for 1D and two-dimensional conditions and with experiments performed in our laboratory for 3D heat transfer using frozen meat. It was applied to predict temperature and water concentration profiles under different thawing conditions in meat products and to simulate the effect of a fat layer located at the surface of the meat piece on temperature profiles. For different product sizes in rectangular geometry, numerical simulations demonstrated that microwave thawing times were significantly lower in comparison to conventional thawing methods. To prevent overheating during thawing, the combination of continuous microwave power with simultaneous application of air convection and the application of microwave power cycles, using refrigerated air convection with controlled surface temperature, were analyzed.