Crank-Nicolson Finite Difference Method Research Articles

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.

Solution of a One-Dimension Heat Equation Using Higher-Order Finite Difference Methods and Their Stability

One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implicit method, and fourth-order implicit Crank-Nicolson finite difference method. Higher-order schemes have complexity in computing values at the neighboring points to the boundaries. It is required there a specification of the values of field variables at some points exterior to the domain. The complexity was incorporated using Hicks approximation. The convergence and stability analysis was also computed for those higher-order finite difference explicit and implicit methods in case of solving a one dimensional heat equation. The obtained numerical results were compared with exact solutions. It is found that backward time and fourth-order centered space implicit scheme along with Hicks approximation performed well over the other mentioned higher-order approaches.

Hall and ion‐slip effects on MHD natural convective flow past an unbounded vertical porous channel with thermodiffusion

AbstractThe consequences of Soret in addition to Dufour of natural convection heat and mass transfer for the unsteady three‐dimensional boundary layer flow through a perpendicular condition of the existence of viscous dissipation, invariable suction, Hall as well as ion slip consequences into relation. The prevailing partial differential equation is dissolved digitally utilizing the implicit Crank–Nicolson finite difference method. The velocity, temperature, as well as concentration dispensations, is addressed computationally and demonstrated by the graphs. Numerical values of the Nusselt number, skin friction as well as Sherwoods numbers nearby the plate are discussed for a choice of values of substantial parameters and are displayed in a tabular manner. It is noticed that the temperature of the fluid diminishes with higher Prandtl numbers. The resulting velocity diminishes with the growing Hartmann number. Rotation, Soret, and Dufour parameters strengthen the velocity and momentum boundary layer thickness. The velocity intensifies through growing Hall and ion‐slip parameters and the revoke trend is acquired with enhancement in suction parameter.

Simultaneous reconstruction of time-dependent coefficients in the parabolic equation from over-specification conditions

The inverse problem of reconstructing the time-dependent coefficients, along with the temperature in a one-dimensional parabolic equation with initial and Neumann boundary conditions supplemented by non-local integral and boundary specification conditions is, for the first time, numerically solved. This is challenging and interesting nonlinear inverse problem which has many significant applications in various fields of physics and mechanics. From the literature, we already know that this inverse problem has a unique solution. However, the problem is still ill-posed (very slight errors in the additional input may cause relatively significant errors in the output coefficients) by being unstable to noise in the input data. For the numerical realization, we apply the Crank–Nicolson finite difference method (FDM) together with the Tikhonov regularization to find a stable and accurate numerical solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both exact and numerically simulated noisy input data are inverted. 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. This study will be significant to researchers working on computational and mathematical methods for solving inverse coefficients reconstruction problems with applications in heat transfer and porous media.

Determination of a time-dependent potential in the higher-order pseudo-hyperbolic problem

The scope of this paper is to determine the time-dependent potential term numerically in the fourth-order pseudo-hyperbolic equation with initial and boundary conditions from an additional measurement condition. From the literature, we already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. For the numerical realization, we apply the Crank–Nicolson finite difference method combined with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted. 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.

Stability and error analysis for a second-order approximation of 1D nonlocal Schrödinger equation under DtN-type boundary conditions

A second-order Crank-Nicolson finite difference method is designed to solve a 1D nonlocal Schrödinger equation on the whole real axis. We employ an asymptotically compatible scheme to discretize the spatially nonlocal operator, and apply the Crank-Nicolson scheme in time to achieve a fully discrete infinite system. An iterative technique for the second-order matrix difference equation is then developed to obtain Dirichlet-to-Dirichlet (DtD)-type artificial boundary conditions (ABCs) with the application of z z -transform for the resulting fully discrete system. After that, with the aid of discrete nonlocal Green’s first identity, we derive Dirichlet-to-Neumann (DtN)-type ABCs from DtD-type ABCs. The resulting DtN-type ABCs are available to reduce the infinite discrete system to a finite discrete system on a truncated computational domain, and make it possible to perform stability and convergence analysis for the reduced problem. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.

ILT20 – an Upgraded Computer Program for Evaluation of Accelerated Leach Test Data of LLW in the Hungarian NPP Paks

Computer Program ILT15 was earlier developed to accompany a new leach test for solidified radioactive waste forms in the Hungarian NPP Paks. The program was designed to be used as a tool for performing the calculations necessary to analyze leach test data, a modelling program to determine if diffusion is the operating leaching mechanism (and, if not, to indicate other possible mechanisms), and a means to make extrapolations using the diffusion models. The program uses computational algorithm of ASTM C1308 standard. Now we have developed another computational algorithm based on a Partial Differential Equation (PDE) of a cylindrical specimen, solved the PDE by a Crank-Nicolson Finite Difference Method (FDM) and calculated by integration the eluted amount of a given component in time. The new solution more accurate method was integrated into the existing ILT15 computer program and the resulting new ILT20 program is able to calculate the Incremental/Cumulative Fraction Leached IFL/CFL) either by ASTM or PDE numerical solution methods.ILT20 program is written in C++ in the Borland C++ Builder programming environment. A detailed description of application of this upgraded modelling computer program is given.

A robust numerical scheme for singularly perturbed parabolic reaction-diffusion problems via the method of lines

In this paper, we consider one- and two-dimensional singularly perturbed parabolic reaction-diffusion problems. We propose a parameter-uniform numerical scheme to solve these problems. The continuous problem is first discretized in the space variable using a fitted operator finite difference method. The partial differential equation is thus transformed into a system of initial value problems which are then integrated in time with the Crank–Nicolson finite difference method. A convergence analysis shows that the scheme is second-order ε-uniform convergent in space and time. Richardson extrapolation of the space variable results in a fourth order ε-uniform convergence. Numerical experiments on two test examples confirm the theoretical findings.

Identifying an unknown potential term in the fourth-order Boussinesq–Love equation from mass measurement

PurposeThe inverse problem of identifying the time-dependent potential coefficient along with the temperature in the fourth-order Boussinesq–Love equation (BLE) with initial and boundary conditions supplemented by mass measurement is, for the first time, numerically investigated. From the literature, the authors already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.Design/methodology/approachFor the numerical discretization, the authors apply the Crank–Nicolson finite difference method along with the Tikhonov regularization for finding a stable and accurate approximate solution. The resulting nonlinear minimization problem is solved using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.FindingsThe present computational results demonstrate that obtained solutions are stable and accurate.Originality/valueThe inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical identification has been studied yet. Therefore, the main aim of the present work is to undertake the numerical realization. The von Neumann stability analysis is also discussed.

Crank–Nicolson finite difference method for time-fractional coupled KdV equation

A Crank–Nicolson finite difference method is presented to solve a system of time fractional coupled Korteweg–de Vries (KdV) equation, the Grünwald–Letnikov definition is used for the time-fractional derivative. The approximation solutions for some numerical examples by the proposed method clarify that the method is applicable, simple and efficient for such problems.

Numerical Solution to Recover Time-dependent Coefficient and Free Boundary from Nonlocal and Stefan Type Overdetermination Conditions in Heat Equation

This paper investigates the recovery for time-dependent coefficient and free boundary for heat equation. They are considered under mass/energy specification and Stefan conditions. The main issue with this problem is that the solution is unstable and sensitive to small contamination of noise in the input data. The Crank-Nicolson finite difference method (FDM) is utilized to solve the direct problem, whilst the inverse problem is viewed as a nonlinear optimization problem. The latter problem is solved numerically using the routine optimization toolbox lsqnonlin from MATLAB. Consequently, the Tikhonov regularization method is used in order to gain stable solutions. The results were compared with their exact solution and tested via root mean squares error (RMSE). We found that the numerical results are accurate and stable.

Quadratic B‐spline collocation method for time dependent singularly perturbed differential‐difference equation arising in the modeling of neuronalactivity

AbstractIn this paper, we consider a time‐dependent singularly perturbed differential‐difference equation with small shifts arising in the field of neuroscience. The terms containing the delay and advance parameters are approximated by using the Taylor's series expansion. The continuous problem is semi‐discretized using the Crank–Nicolson finite difference method in the time direction on uniform mesh and quadratic B‐spline collocation method in the space direction on exponentially graded mesh. The method is shown to be second‐order uniformly convergent in space and time direction. Theoretical estimates are carried out which support the obtained numerical experiments.

A note on stochastic polynomial chaos expansions for uncertain volatility and Asian option pricing

This paper concerns accurate and efficient polynomial chaos expansions (PCEs) for Asian option pricing with uncertain volatilities. While arbitrary distributions of the volatility parameter are applied for estimating real-world option prices, arbitrary polynomial chaos (aPC) are incorporated for approximating raw data of the historical volatility distributions. Rigorous analysis is carried out to ensure the numerical stability of the compact aPC Crank-Nicolson finite difference method accomplished. Numerical results acquired are compared with solutions via standard Monte Carlo schemes (MCSs) and generalized polynomial chaos (gPC) with different random volatilities. Stock data from Asian financial industry are used. It is evident that the novel schemes derived are highly accurate and efficient for evaluating means and variances of uncertain volatility and stochastic Asian option pricing.

Predicting foreign currency exchange rates using the numerical solution of the incompressible Navier–Stokes equations

The movement of exchange rates has similarities with physical phenomena, especially the movement of fluid in turbulence. The parameters change of the data information on the foreign exchange market will result in unstable exchange rates movement. These are according to changes in energy in the turbulence flow resulting in fluid particles moving randomly at fluctuating velocity. This study will predict the exchange rates of the United States Dollar (USD) to Indonesian Rupiah (IDR) using the incompressible Navier–Stokes equations which have been modified with economic parameters. The analogy of economic parameters to physical parameters such as Gross Domestic Product (GDP) as the distance between the two measured velocity points, the trade balance of two countries as fluid viscosity, and fluid flow velocity as exchange rate changes, while the inflation rates and interest rates are external forces. The numerical solution of the Navier–Stokes equations is solved using the Crank–Nicolson finite difference method. The results of the prediction of the exchange rate of USD to IDR are compared with actual exchange rate data. The exchange rate prediction results of the Navier–Stokes equations that calculated using the Crank–Nicolson scheme for one year produce an error percentage below 2.5 % of the actual exchange rate data.

Thermal extraction of water ice from the lunar surface - A 3D numerical model

We have used a three-dimensional thermal and gas diffusion model to study the heated extraction of water ice from lunar regolith. Both surface heating and the insertion of heated drills were investigated for various heating power levels and expected soil ice fractions. Our calculations rely on the use of the Crank-Nicolson finite difference method for the diffusion equations.We find that the extraction of water vapor from lunar regolith requires high levels of heating power in the investigated configurations. Any heating below about 1000 ​W per m2 or per heated drill produces negligible amounts of vapor. We also find that extraction efficiencies are highly dependent on the heating configuration, as well as the initial ice fraction present in the regolith. In addition, we find that, depending on the heating configuration, significant amounts of water escape the heated volume and are lost through refreeze in colder regions.We conclude that water extraction through heating will be a power-hungry process if not optimized through additional controls (e.g. volume confinement by heated walls). In addition, the choice of a heating configuration for optimal extraction efficiency depends on the ice content of the soil. Therefore, detailed prospecting will be essential to the design of future ISRU activities on the Moon.

A Comparative Study between Implicit and Crank-Nicolson Finite Difference Method for Option Pricing

In this paper, we have applied the finite difference methods (FDMs) for the valuation of European put option (EPO). We have mainly focused the application of Implicit finite difference method (IFDM) and Crank-Nicolson finite difference method (CNFDM) for option pricing. Both these techniques are used to discretized Black-Scholes (BS) partial differential equation (PDE). We have also compared the convergence of the IFDM and CNFDM to the analytic BS price of the option. This turns out a conclusion that both these techniques are fairly fruitful and excellent for option pricing.
 GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 13-27

A Crank-Nicolson Approximation for the time Fractional Burgers Equation

Abstract In the present manuscript, Crank Nicolson finite difference method is going to be applied to get the approximate solutions for the fractional Burgers equation. The fractional derivative used in this equation is going to be taken into consideration in the Caputo sense. The L1 type discretization formula is going to be applied to this equation. For checking the efficiency of proposed methods, the error norms L 2 and L ∞ have at the same time been calculated. Those newly got solutions using the presented method illustrate the easy usage and efficiency of the approach presented in this manuscript.