Crank-Nicolson Type Scheme Research Articles

Variable thermal conductivity and mass diffusivity effects in a free convective flow of doubly stratified non-darcian porous medium over a vertical plate

The research in this article is carried out to study incompressible and unsteady free convective flow on a semi-infinite isothermal vertical plate in a doubly stratified non-Darcian porous media with variable mass diffusivity and variable thermal conductivity. The governing non-linear partial differential equations of flow were calculated by applying an implicit finite difference scheme of Crank-Nicolson type. Various parametric impacts on concentration profiles, temperature, velocity, as well Sherwood number, Nusselt number and skin friction, were examined and presented in graphs. It is examined that there exists a significant temperature decrease for high Darcy number in stratified fluids. Also, it is detected that the presence of stratification produces a considerable drop in skin friction while increases the mass and heat transfer rate. Comparison of current outcomes well agreed with the available solutions.

Christov expansion method for nonlocal nonlinear evolution equations

Christov functions are a complete orthonormal set of functions on L 2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves.

Binary thermal fluids computation over arbitrary surfaces with second-order accuracy and unconditional energy stability based on phase-field model

In this paper, we propose an efficient surface computational system with second-order spatial and temporal accuracy to solve multiple physical field coupling problems over arbitrary surfaces. The computational system is coupled with heat transfer equation and incompressible Navier–Stokes equation based on a phase-field model. Due to the discretization of the triangular grids, we define the discretized gradient operator, divergence operator and the Laplace–Beltrami operator with second-order spatial accuracy. The Crank–Nicolson-type scheme is used to confirm the temporal accuracy. The Navier–Stokes equation is solved by the projection method. We use the biconjugate gradient stabilized method to solve the involved equations. The discrete system is provable to be unconditionally stable and the mass conservation law is satisfied during the computation, which implies that the proposed method is not limited by the temporal step. Several computational tests are conducted to show the efficiency, robustness and accuracy of the proposed scheme.

An unconditionally energy stable method for binary incompressible heat conductive fluids based on the phase–field model

This paper proposes an unconditionally energy stable method for incompressible heat conductive fluids under the phase–field framework. We combine the complicated system by the Navier–Stokes equation, Cahn–Hilliard equation, and heat transfer equation. A Crank–Nicolson type scheme is employed to discretize the governing equation with the second-order temporal accuracy. The unconditional energy stability of the proposed scheme is proved, which means that a significantly larger time step can be used. The Crank–Nicolson type discrete framework is applied to obtain the second-order temporal accuracy. We perform the biconjugate gradient method and Fourier transform method to solve the discrete system. Several computational tests are performed to show the efficiency and robustness of the proposed method.

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

In this study, our attention is focused on deriving integrals of motion (conservation laws; invariants) for the problem of an optical pulse propagation in an optical fiber containing an optical amplifier or attenuator because, to date, such invariants are absent in the literature. The knowledge of a problem’s invariants allows us develop finite-difference schemes possessing the conservativeness property, which is crucial for solving nonlinear problems. Laser pulse propagation is governed by the nonlinear Ginzburg–Landau equation. Firstly, the problem’s conservation laws are developed for the various parameters’ relations: for a linear case, for a nonlinear case without considering the linear absorption, and for a nonlinear case accounting for the linear absorption and homogeneous shift of the pulse’s phase. Hereafter, the Crank–Nicolson-type scheme is constructed for the problem difference approximation. To demonstrate the conservativeness of the constructed implicit finite-difference scheme in the sense of preserving difference analogs of the problem’s invariants, the corresponding theorems are formulated and proved. The problem of the finite-difference scheme’s nonlinearity is solved by means of an iterative process. Finally, several numerical examples are presented to support the theoretical results.

Fully-discrete energy-preserving scheme for the space-fractional Klein–Gordon equation via Lagrange multiplier type scalar auxiliary variable approach

A family of effective fully-discrete energy-preserving schemes for the space-fractional Klein–Gordon equation is developed in this paper. First, the recently developed Lagrange multiplier type scalar auxiliary variable approach is employed to obtain a new equivalent system from the original space-fractional Klein–Gordon system. Then, a family of special second-order implicit, explicit and implicit approximations to respectively discretize the linear parts, nonlinear parts and time-derivative parts are obtained in the above equivalent system to establish a family of semi-discrete (continuous in space) energy-preserving schemes. Furthermore, the Fourier pseudo-spectral method is used to discretize the space for extending to the fully-discrete case and rigorous theoretical proofs guarantee its conservation of original energy. Especially, the well-known implicit–explicit Crank–Nicolson type scheme is only one of the above-mentioned schemes. It is inspiring that the main computational efforts of this method in each time step are only to solve two linear, decoupled differential equations with constant coefficients different from non-homogeneous terms, which thus can be effectively solved. Finally, numerical experiments are carried out to verify the theoretical results of the accuracy, efficiency and conservation of original energy.

Transient free convection heat and mass transfer of Casson nanofluid over a vertical porous plate subjected to magnetic field and thermal radiation

In this present study, the transient magnetohydrodynamics free convection heat and mass transfer of Casson nanofluid past an isothermal vertical flat plate embedded in a porous media under the influence of thermal radiation is studied. The governing systems of nonlinear partial differential equations of the flow, heat and mass transfer processes are solved using implicit finite difference scheme of Crank-Nicolson type. The numerical solutions are used to carry out parametric studies. The temperature as well as the concentration of the fluid increase as the Casson fluid and radiation parameters as well as Prandtl and Schmidt numbers increase. The increase in the Grashof number, radiation, buoyancy ratio and flow medium porosity parameters causes the velocity of the fluid to increase. However, the Casson fluid parameter, buoyancy ratio parameter, the Hartmann (magnetic field parameter), Schmidt and Prandtl numbers decrease as the velocity of the flow increases. The time to reach the steady state concentration, the transient velocity, Nusselt number and the local skin-friction decrease as the buoyancy ratio parameter and Schmidt number increase. Also, the steady-state temperature and velocity decrease as the buoyancy ratio parameter and Schmidt number increase. Also, the local skin friction, Nusselt and Sherwood numbers decrease as the Schmidt number increases. However, the local Nusselt number increases as the buoyancy ratio parameter increases. It was established that near the leading edge of the plate), the local Nusselt number is not affected by both buoyancy ratio parameter and Schmidt number. It could be stated that the present study will enhance the understanding of transient free convection flow problems under the influence of thermal radiation and mass transfer as applied in various engineering processes.

The Numerical Solution of Heat Equation by the Fourth-Order Iterative Alternating Decomposition Explicit Method with MPI

The numerical solution of the heat equation in one space dimension is obtained using the Fourth-Order Iterative Alternating Decomposition Explicit Method (4-IADE) on a parallel platform with Message Passing Interface (MPI). Here, a higher fourth-order Crank-Nicolson type scheme is used in the approximation which gives rise to a Penta diagonal matrix in the solution of the system at each time level. The method employs a splitting strategy which is applied alternately at each half time step. The method is shown to be computationally stable and appropriate parameters chosen to accelerate convergence. The accuracy of the method is comparable to that of existing well known methods. Results obtained by this method for several different problems were compared with the exact solution and agreed closely with those obtained by other finite-difference methods with correlation between speedup and efficiency.

A New Second Order Approximation for Fractional Derivatives with Applications

We propose a generalized theory to construct higher order Grünwald type approximations for fractional derivatives. We use this generalization to simplify the proofs of orders for existing approximation forms for the fractional derivative. We also construct a set of higher order Grünwald type approximations for fractional derivatives in terms of a general real sequence and its generating function. From this, a second order approximation with shift is shown to be useful in approximating steady state problems and time dependent fractional diffusion problems. Stability and convergence for a Crank-Nicolson type scheme for this second order approximation are analyzed and are supported by numerical results.

Flow visualization using heat lines for unsteady radiative hydromagnetic micropolar convection from a vertical slender hollow cylinder

The present study aims to investigate the thermal radiation heat transfer effect on unsteady magnetohydrodynamic flow of micropolar fluid over a uniformly heated vertical hollow cylinder using Bejan's heat function concept. The normalized conservation equations emerge as a system of time-dependent non-linear coupled partial differential equations. Under appropriate wall and free stream conditions these equations are solved with an efficient unconditionally stable implicit scheme of Crank-Nicolson type. Important thermo-physical parameters featured include the magnetic body force parameter (M), Grashof (free convection) parameter (Gr), Eringen micropolar material parameter (K), Prandtl number (Pr), conjugate heat transfer parameter (P) and radiative-conductive Rosseland parameter (N), are analyzed on the flow-field with ranges 0–3, 105–106, 0–1.2, 0.7–7.0, 0–0.5 and 0–15, respectively. The time-histories of average values of momentum and heat transport coefficients, as well as the steady-state flow variables are presented for selected values of these non-dimensional parameters. With elevation in magnetic parameter or radiation parameter, the time taken for the flow-field variables to attain the time-independent state increases. The dimensionless thermal radiative heat function values are closely correlated with the overall rate of heat transfer on the outer hot cylindrical wall. Bejan's heat flow visualization implies that the thermal radiative heat function contours are compact in the neighbourhood of leading edge of the boundary layer on the outer hot cylindrical wall. Increasing radiation or magnetic parameter values result in an increase in the deviation of heat lines from the hot wall. Also, the heat lines are observed to depart slightly away from the hot wall with greater values of vortex viscosity. Furthermore, the deviations of flow variables from the hot wall for a micropolar fluid are significant compared to the Newtonian fluid (vanishing micropolar vortex viscosity).

A class of two-level implicit unconditionally stable methods for a fourth order parabolic equation

In this paper, we discuss two-level implicit methods for the solution of a special type of fourth order parabolic partial differential equation of the form uxxxx-2uxxt+utt=f(x,t,u), 0<x<1, t>0 subject to appropriate initial and Dirichlet boundary conditions by converting the original problem to a coupled system of two second order parabolic equations. We use only three spatial grid points and it is not required to discretize the boundary conditions. The proposed Crank-Nicolson type scheme is second order accurate in both the temporal and spatial dimensions while the compact Crandall's type scheme is second order accurate in temporal and fourth order accurate in spatial dimension. The methods do not require any fictitious nodes outside the solution domain for handling the boundary conditions. For a fixed mesh ratio parameter (∆t/∆x2), the proposed Crandall's type method behaves like a fourth order method in space. Using matrix stability analysis, the proposed methods are shown to be unconditionally stable. The resulting implicit difference formulas gives block tri-diaginal matrix structure which is solved efficiently using block Gauss-Seidel method or block Newton method depending on linear or nonlinear behaviour of the equations. The methods compute the numerical value of u and time-dependent Laplacian uxx−ut, simultaneously. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed methods.

A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions

In this article we claim that we are going to give a priori and a posteriori error estimates for a Crank Nicolson type scheme. The problem is discretized by the finite elements in space. The main result of this paper consists in establishing two types of error indicators, the first one linked to the time discretization and the second one to the space discretization.

Unsteady Natural Convective Flow over an Impulsively Started Semi-Infinite Vertical Plate in the Presence of Porous Medium with Chemical Reaction

An investigation is carried out to analyze the effects of heat and mass transfer over an impulsively started vertical plate in the presence of porous medium with chemical reaction. The unsteady, non-linear, coupled partial differential equations are solved by implicit finite difference scheme of Crank Nicolson type. The influence of various parameters like Prandtl number, Schmidt number, first order chemical reaction on the velocity, temperature and concentration are analyzed. The local skin friction, local Nusselt number, local Sherwood number, average skin friction, average Nusselt number and average Sherwood numbers are investigated. It is observed that the velocity and concentration boundary layer decreases with increasing chemical reaction. An increase in the Schmidt number reduces the concentration boundary layer thickness.

Transient magnetohydrodynamic flow of Eyring-Powell fluid in a porous medium

This study investigated the effects of magnetic field, thermal radiation and viscous dissipation on transient magneto hydrodynamic (MHD) flow of non-Newtonian incompressible fluid obeying Eyring-Powell model in a porous medium. The governing equations were formulated and transformed into non-dimensional equations. Numerical solution to the transformed governing nonlinear partial differential equations was obtained using the implicit finite difference scheme of Crank-Nicolson type. The finite difference equations form Thomas algorithm tri-diagonal matrix system of equations which was solved using the MATLAB. The results showed that a rise in Non-Newtonian parameter F, thermal Grashof number Gr, modified Grashof number Gm and dissipation function Ec, caused velocity to increase whereas velocity decreased with increase in Non-Newtonian parameter A, magnetic field parameter M, radiation parameter R, Schmidt number Sc, Prandtl Number Pr and chemical reaction parameter, g. Temperature increased with increase in dissipation function Ec, while it decreased as Prandtl number, magnetic field parameter and Radiation number increased. Increase in Schmidt number and chemical reaction result to a decrease in concentration. Keywords : Non-Newtonian; Radiation; Magnetic Field; Eyring-Powell Model; Dissipation.

Unsteady Magnetohydrodynamic Casson Fluid Flow over a Vertical Cone and Flat Plate with Non-Uniform Heat Source/Sink

The present study focuses the effects of double dispersion, non-uniform heat source/sink and higher order chemical reaction on unsteady, free convective, MHD Casson fluid flow over a vertical cone and flat plate saturated with porous medium. The extensively validated and unconditionally stable numerical solutions are obtained for the governing equations of two dimensional boundary layer flow by using the finite difference scheme of Crank-Nicolson type. The behavior of velocity, temperature and concentration distributions for various controlling parameters of this problem are graphically illustrated and discussed in detail. The average skin friction, Nusselt number and Sherwood number for sundry parameters are presented in tables. Results indicate that an increase in Casson fluid parameter is found to decelerate fluid flow by increasing the plastic dynamic viscosity whereas it enhances the shear stress in the flow regime. The temperature-dependent heat source/sink plays a vital role on controlling the heat transfer however the surface-dependent heat source/sink also has notable influence on the heat transfer characteristics. It is to be noted that higher order chemical reaction has the tendency to dilute the influence of chemical reaction parameter on the species concentration.

Brinkman Model for Unsteady MHD free Convection Flow of a Chemical Reacting Doubly Stratified Fluid with Heat Generation and Radiation

In this investigation, the effects of magneto-hydrodynamics, heat generation and thermal radiation have been discussed on unsteady natural convection flow in a Brinkman porous medium saturated with a chemically reacting doubly stratified fluid. Initially, the flow is described through time-dependent differential equations. These are converted into a set of coupled non-dimensional differential equations. Crank-Nicolson type scheme is employed to analyze the solution of the model. In order to analyze the usefulness of the investigation the dimensionless local skin-friction, heat and mass transfer rates have been presented graphically for different combinations of flow governing parameters.

A Crank-Nicolson Scheme for the Dirichlet-to-Neumann Semigroup

The aim of this work is to study a semidiscrete Crank-Nicolson type scheme in order to approximate numerically the Dirichlet-to-Neumann semigroup. We construct an approximating family of operators for the Dirichlet-to-Neumann semigroup, which satisfies the assumptions of Chernoff’s product formula, and consequently the Crank-Nicolson scheme converges to the exact solution. Finally, we write aP1finite element scheme for the problem, and we illustrate this convergence by means of a FreeFem++ implementation.

Effects of Cross-Diffusion on Unsteady Natural Convection past a Vertical Flat Plate in an Electrically Conducting Doubly Stratified Fluid Saturated Brinkman Porous Medium

In this paper, the influence of viscous dissipation and cross-diffusion effects on unsteady natural convective flow of an electrically conducting doubly stratified fluid in a Brinkman porous medium has been analyzed. By using the similarity transformations, the governing differential equations are expressed into a set of non-linear coupled ordinary differential equations along with the suitable boundary conditions. The Crank-Nicolson type scheme developed based on implicit finite difference method has been used to solve the reduced nonlinear boundary value problem. Numerical results for local skin-friction, local Nusselt number, and local Sherwood number profiles are depicted graphically and analyzed in detail for different physical parameters.

Heat and Mass Transfer Effects on Unsteady MHD Radiative Flow of a Chemically Reacting Fluid Past an Impulsively Started Vertical Plate

The interaction of free convection with thermal radiation of viscous in- compressible MHD unsteady chemical reacting fluid flow past an impulsively started vertical plate is analyzed. The Rosseland approximation is used to describe the ra- diative heat transfer in the limit of the optically thin fluid. The non-linear, coupled equations are solved using an implicit finite difference scheme of Crank-Nicolson type. Velocity, temperature and concentration of the flow have been presented for various parameters such as thermal Grashof number, mass Grashof number, Prandtl number, Schmidt number, radiation parameter and magnetic parameter. The local and average skin friction, Nusslet number and Sherwood number are also presented graphically. It is observed that, when the radiation parameter increases the velocity and temperature decrease in the boundary layer. Also, it is found that the presence of chemical reac- tion parameter leads to decrease in the velocity field and concentration and rise in the thermal boundary thickness. Keywords : Heat transfer; MHD; Radiation; Finite-difference Scheme; Vertical plate 2010 Mathematics Subject Classification 74A15

Application of Richardson extrapolation for multi-dimensional advection equations

A Crank–Nicolson type scheme, which is of order two with respect to all independent variables, is used in the numerical solution of multi-dimensional advection equations. Normally, the order of accuracy of any numerical scheme can be increased by one when the well-known Richardson Extrapolation is used. It is proved that in this particular case the order of accuracy of the combined numerical method, the method consisting of the Crank–Nicolson scheme and the Richardson Extrapolation, is not three but four.