Crank-Nicolson Finite Difference Scheme Research Articles

Development of a Model for Analysing Radial Flow of Slightly Compressible Fluids

In this work, we develop a finite element-finite difference method to solve the differential equation governing the radial flow of slightly compressible fluids. The finite element method is used to carry out spatial approximations so as to study the variation of fluid properties at the various nodes to which effect we divided the entire radial domain of the fluid into a mesh of four radial 1-D quadratic elements which exposes nine nodes to intense study. Time approximation is done with the aid of the Crank-Nicolson finite difference scheme.

Inverse space-dependent perfusion coefficient identification

The identification of the space-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. In this inverse coefficient identification problem, the additional measurement necessary to render a unique solution is a boundary temperature measurement. A numerical approach based on a Crank-Nicolson finite-difference scheme combined with Tikhonov's regularization methods is developed. Numerical results are presented and discussed.

Crank–Nicolson finite difference scheme for the Rosenau–Burgers equation

In this paper, a Crank–Nicolson finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau–Burgers equation is proposed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is convergent in the order of o ( τ 2 + h 2 ) and stable. Numerical simulations show that the method is efficient.

A Mathematical Analysis of Intravitreal Drug Transport

Purpose: The aim of our present work is the development of a quasi steady-state model for the distribution of intravitreally injected drugs and investigation of the effects of various model parameters on the drug distribution in normal and diseased eyes. Method: A simple mathematical model for the intravitreal transport of drugs was developed using Fick's law of diffusion, Darcy's law of convective flow, and Michaelis –Menten kinetics of metabolism. A Crank- Nicolson finite difference scheme of the equation describing the drug transport in the vitreous body was written, in which the radial and axial diffusive terms and convective terms of the equation were approximated by central differences, while the temporal terms were approximated by average of forward and backward time differences. A system of linear algebraic equations obtained from the Crank Nicolson finite difference scheme was solved by line Jacobi iterative scheme in which successive improved approximate results are obtained. Result: The model predicts that an increase in the metabolic (consumption) rate and drug release rate constant reduces the concentration of intravitreally injected drug at the centre of retina and along centreline of vitreous body. A significant increase in the drug concentration at the centre of retina and along the centreline of the vitreous body in the eye afflicted with glaucoma and retinal detachment is observed and the decay rate of drug concentration in these pathological states is higher than that in the normal eyes. Conclusion: The vitreous outflow as observed in the glaucomatous and/- or rhegmatogenous eyes may contribute to the transport of intravitreally injected drug in the vitreous body. The drug concentration in the vitreous body and at the centre of retina in such diseased eyes is higher than that in the normal eyes and the decay rate of drug concentration is significantly enhanced. Keywords: Convective-diffusive transport, intravitreal injection, line-Jacobi iterative technique, release rate.Tropical Journal of Pharmaceutical Research Vol. 7 (1) 2008: pp. 867-877

Transparent boundary conditions for quantum-waveguide simulations

We consider the two-dimensional, time-dependent Schrödinger equation discretized with the Crank–Nicolson finite difference scheme. For this difference equation we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. We apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.

Coupling stabilized finite element methods with finite difference time integration for advection–diffusion–reaction problems

We present some numerical schemes for the unsteady advection–diffusion–reaction linear problem, in one space dimension. We investigate two possible different ways of combining the discretization in time and in space (where the sequence of the discretizations is interchanged). Discretization in time is performed by using the Crank–Nicolson finite difference scheme, while for the space discretization we consider three classical stabilized finite element schemes and the more recent Link-Cutting Bubble strategy proposed in [F. Brezzi, G. Hauke, L.D. Marini, G. Sangalli, Link-cutting bubbles for the stabilization of convection–diffusion–reaction problems, Math. Models Methods Appl. Sci. 13 (2003) 445–461]. Numerical experiments are presented to assess and valuate the capabilities of the proposed methods. An L 1-error analysis of the Link-Cutting Bubble strategy for solving the steady problem is included.

On the numerical solution of radioactivity migration in a porous medium

The deep burial of vitrified radioactive waste in rock and the storage of spent fuel rods in storage tanks at sufficient depths call for the modeling of the migration of radioactivity in rock. Rock is considered as a system of connected parallel fractures. The two coupled partial differential equations which govern the transport in the fracture and in the porous matrix need to be solved by an efficient numerical procedure. Here, we indicate an accurate scheme that has the following two components. The first is the use of a very high order derivative approximation for the concentration gradient at the interface of the waste-matrix and the fracture. The second is the use of Crank–Nicolson finite difference scheme and this gives numerical values which are second order accurate both in space and time variables. With this prescription, reliable estimates of the concentration can be obtained at a distance of 500 m from the source for both long lived and short lived species. Typical concentration plots for few important species are provided.

On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part II

We consider initial-boundary value problems for a generalized time-dependent Schrodinger equation in $1D$ on the semi-axis and in $2D$ on a semi-bounded strip. For Crank-Nicolson finite-difference schemes, we suggest an alternative coupling to approximate transparent boundary conditions and present a condition ensuring unconditional stability. In the case of discrete transparent boundary conditions, we revisit the statement and the proof of stability together with the derivation of the conditions.

Lattice Boltzmann schemes for the nonlinear Schrödinger equation

The lattice Boltzmann (LB) method is applied to solve the time-dependent nonlinear Schrödinger (NLS) equation. Through approximating the reaction term at different orders of accuracy, three diffusion-reaction LB schemes are constructed for the cubic NLS equation. A LB initial condition is proposed to include the first-order nonequilibrium distribution function. These LB schemes are used to solve the one-soliton propagation and the homoclinic orbit problems. Detailed simulation results confirm that the high-order reaction term and the LB initial condition are effective in reducing the truncation errors. Compared with the Crank-Nicolson finite difference scheme, the LB scheme is found to give at least comparable and generally more accurate approximation for the cubic NLS equation.

A numerical method based on Crank-Nicolson scheme for Burgers’ equation

In this paper, we present a solution based on Crank-Nicolson finite difference method for one-dimensional Burgers’ equation. Burgers’ equation arises frequently in mathematical modeling of problems in fluid dynamics. Hopf-Cole transformation [E. Hopf, The partial differential equation u t + uu x = νu xx , Commun. Pure Appl. Math. 3 (1950) 201–230, J.D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951) 225–236] is used to linearize Burgers’ equation, the resulting heat equation is discretized by using Crank-Nicolson finite difference scheme. This method is shown to be unconditionally stable and second order accurate in space and time. Numerical results obtained by the present method have been compared with exact solution for different values of Reynolds’ number.

Modeling Fate and Transport of Roadside-Applied Herbicides

A physically based, one-dimensional, mathematical model is developed to simulate transport of herbicides applied to roadsides into adjacent surface water. Herbicide transport by overland flow, infiltrating water, and subsurface flow is considered. The model accounts for advection, dispersion, equilibrium linear sorption in the soil, and first-order decay of the herbicides. Chemical transport between soil and overland flow is described using a rate-limited mass exchange between the two compartments that is proportional to the difference between the chemical concentration in overland flow and the pore water with a proportionality constant that is a function of overland flow characteristics and rain intensity. For simulating overland and subsurface flow, the kinematic wave equation is solved using a Crank-Nicolson finite difference scheme. Richards' equation is solved using a Galerkin finite element method to model infiltration, and the transport equation is solved using a finite difference method. The model is calibrated and verified using extensive field data on the runoff of five herbicides with a range of physical-chemical properties at two field sites. Good agreement is obtained between predicted and measured event-mean concentrations in runoff and the herbicide concentration remaining in the application-zone soil.

A Model for CD2/CD58-Mediated Adhesion Strengthening

Stable cell adhesion is vital for structural integrity and functional efficacy. Yet how low affinity adhesion molecules such as CD2 and CD58 can produce stable cell adhesion is still not completely understood. In this paper, we present a theoretical model that simulates the accumulation of CD2 and CD58 in the contact area of a Jurkat T lymphoblast and a CD58-containing substrate. The cell is assumed to have a spherical shape initially and it is allowed to spread gradually on a circular substrate. Mobile CD2 and CD58 can diffuse freely on both the cell and substrate. Their binding in the contact area is controlled by first-order kinetics. The contact area grows linearly with the total number of CD2/CD58 bonds. Cellular deformation and cytoskeleton involvement were not considered. This time-dependent moving-boundary problem was solved with the Crank-Nicolson finite difference scheme and the variable space grid method. Our simulated results are in reasonable agreement with the experimental observations. The role of diffusion becomes more and more prominent during the contact area increase, which is not sensitive to the kinetic rate constants tested in this study. However, it is very sensitive to the dissociation equilibrium constant and the concentrations of CD2 and CD58.

Evaluation of double average asian options by the legendre spectral method

In this paper, Hie evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were discussed as examples. The problem is a parabolic one on a finite domain whose equation degenerates into ordinary differential equations on the boundaries. A fully discrete scheme was established by using the Legendre spectral method in space and the Crank-Nicolson finite difference scheme in time. The stability and convergence of the scheme were analyzed. Numerical results show that the method can keep the spectral accuracy in space for such degenerate problems.

Initial Boundary-Value Problems for Derivative Nonlinear Schroedinger Equation. Justification of Two-Step Algorithm

We investigate two different initial boundary-value problems for derivative nonlinear Schrödinger equation. The boundary conditions are Dirichlet or generalized periodic ones. We propose a two-step algorithm for numerical solving of this problem. The method consists of Bäcklund type transformations and difference scheme. We prove the convergence and stability in C and H1 norms of Crank–Nicolson finite difference scheme for the transformed problem. There are no restrictions between space and time grid steps. For the derivative nonlinear Schrödinger equation, the proposed numerical algorithm converges and is stable in C1 norm.

An Electrochemical Model of the Transport of Charged Molecules through the Capillary Glycocalyx

An electrochemical theory of the glycocalyx surface layer on capillary endothelial cells is developed as a model to study the electrochemical dynamics of anionic molecular transport within capillaries. Combining a constitutive relationship for electrochemical transport, derived from Fick's and Ohm's laws, with the conservation of mass and Gauss's law from electrostatics, a system of three nonlinear, coupled, second-order, partial, integro-differential equations is obtained for the concentrations of the diffusing anionic molecules and the cations and anions in the blood. With the exception of small departures from electroneutrality that arise locally near the apical region of the glycocalyx, the model assumes that cations in the blood counterbalance the fixed negative charges bound to the macromolecular matrix of the glycocalyx in equilibrium. In the presence of anionic molecular tracers injected into the capillary lumen, the model predicts the size- and charge-dependent electrophoretic mobility of ions and tracers within the layer. In particular, the model predicts that anionic molecules are excluded from the glycocalyx at equilibrium and that the extent of this exclusion, which increases with increasing tracer and/or glycocalyx electronegativity, is a fundamental determinant of anionic molecular transport through the layer. The model equations were integrated numerically using a Crank-Nicolson finite-difference scheme and Newton-Raphson iteration. When the concentration of the anionic molecular tracer is small compared with the concentration of ions in the blood, a linearized version of the model can be obtained and solved as an eigenvalue problem. The results of the linear and nonlinear models were found to be in good agreement for this physiologically important case. Furthermore, if the fixed-charge density of the glycocalyx is of the order of the concentration of ions in the blood, or larger, or if the magnitude of the anionic molecular valence is large, a closed-form asymptotic solution for the diffusion time can be obtained from the eigenvalue problem that compares favorably with the numerical solution. In either case, if leakage of anionic molecules out of the capillary occurs, diffusion time is seen to vary exponentially with anionic valence and in inverse proportion to the steady-state anionic tracer concentration in the layer relative to the lumen. These findings suggest several methods for obtaining an estimate of the glycocalyx fixed-charge density in vivo.

Effect of heat transfer direction on the numerical prediction of beef freezing processes

Accurate numerical prediction of freezing processes for meats is difficult. One of the reasons is the effect of heat transfer direction during freezing. When meat is being frozen, heat can transfer parallel to or perpendicular to muscle fibres of meats at various rates. In this study the influence of the heat transfer direction on the numerical simulation of the freezing processes was analysed. A computer simulation program using the modified Crank–Nicolson finite difference scheme was developed to predict the temperature profiles during freezing. Two thermal conductivity formulas, the Levy and the modified series equations, were incorporated into the program. Experiments were carried out on beef samples. The comparison between the numerical results and experimental data shows that the program can predict the freezing processes well.

Linear and nonlinear wave diffraction using the nonlinear time dependent mild slope equations

Nonlinear wave diffraction is studied using the nonlinear time-dependent mild slope equation. The equations are solved using a combined Newton–Raphson and Crank–Nicolson finite difference scheme. The model results are verified for propagation of highly nonlinear waves over uniform depth and wave diffraction due to semi-finite breakwater and breakwater gap with different widths. Comparison between the numerical and experimental results indicates that the model is capable of simulating nonlinear wave diffraction. The model is applied to study the effect of the wave nonlinearity on the diffraction coefficient for a semi-infinite breakwater and a breakwater gap.

The impulsively started flow over oblate spheroids

The problem of the impulsively started flow over an oblate spheroid is solved using the series truncation method in which the stream function and vorticity are expanded in terms of series of Legendre functions. The resulting time-dependent differential equations are solved using the Crank-Nicolson finite difference scheme. The parameters involved are the Reynolds number and the axis ratio. The range of Reynolds numbers considered is from 5 to 100 while the axis ratio is considered at values of A r = 0.6 and 0.76. The time variation of the flow field is presented in the form of streamline and equi-vorticity patterns as well as pressure and surface vorticity distributions. The time development of the separation angle, wake length, and the frictional and total drag coefficients are also presented. The large time (steady) values of the drag coefficient for the special case of a sphere are compared with previously known data and the agreement is satisfactory.

Solution and Evaluation of Permafrost Thaw-Subsidence Model

In this paper, we develop a numerical solution to equations derived in the first part of this study to simulate the thaw subsidence of an unsaturated frozen soil due to surface heating. Solutions obtained in terms of pore pressure, temperature, degree of unfrozen water content, ice content, and porosity provide the spatial and temporal variations of these quantities in a thawing soil column. A one-dimensional form of governing equations is discretized by employing a Crank-Nicolson finite-difference scheme. These highly nonlinear equations are then solved by an iterative procedure at every time step for predescribed positive surface temperatures that produce thawing. The numerical solutions are obtained for a 1-m silt column. Results show that, above the frozen region, unfrozen water saturation increases due to low hydraulic conductivity, which prevents the melting ice from draining, thus causing accumulation of water above it. It has also been observed that most of the settlement at the surface is due to thaw consolidation rather than settlement due to drainage. The former is completed in seven days of a 12-day simulation period. Furthermore, a sensitivity analysis was carried out to determine the response of the simulation to deviations in the values of certain parameters. It is found that the settlement estimations are very sensitive to the thaw-settlement parameter and phase-composition curve, thus requiring accurate determination of these parameters for a reliable simulation of the thaw-consolidation process.

Finite difference solutions for parabolic equations with the time weighting initial conditions

A class of nonlinear parabolic boundary value problems with time weighting initial condition is considered. Instead of studying the existence, uniqueness, and continuous dependence of the solution upon the data we give a brief review of the known results and then concentrate on the performance of the backward Euler and Crank-Nicolson finite difference schemes [1]. The results of some numerical examples are presented which demonstrate the efficiency and rapid convergence.