Implicit Crank-Nicolson Method Research Articles

EFFICIENT NUMERICAL METHODS FOR THE KDV EQUATION

We consider the second order Strang splitting method to approximate the solution to the KdV equation. The model equation is split into three sets of initial value problems containing convection and dispersal terms separately. TVD MUSCL or MUSCL scheme is applied to approximate the convection term and the second order centered difference method to approximate the dispersal term. In time stepping, explicit third order Runge-Kutta method is used to the equation containing convection term and implicit Crank-Nicolson method to the equation containing dispersal term to reduce the CFL restriction. Several numerical examples of weakly and strongly dispersive problems, which produce solitons or dispersive shock waves, or may show instabilities of the solution, are presented.

Numerical analysis for migration of austenite/ferrite interface during carburization of Fe

During isothermal annealing of the ferrite of a binary Fe–C alloy in an appropriate carburization atmosphere at temperatures of 1020–1180 K, the austenite will be formed as a layer on the surface of the ferrite. The migration behavior of the austenite/ferrite interface during the isothermal carburization was numerically analyzed using the diffusion equation describing the flux balance at the interface. However, the diffusion coefficient of C in the austenite monotonically increases with increasing concentration of C. Thus, the composition dependence of the diffusion coefficient was taken into consideration. For the numerical analysis, Crank–Nicolson implicit method was combined with a finite-difference technique. According to the numerical calculation, the parabolic relationship holds between the migration distance of the austenite/ferrite interface and the annealing time. The composition dependence of the diffusion coefficient yields acceleration of the migration. The numerical calculation satisfactorily reproduces the experimental result of the diffusion controlling migration.

WAF Method and Splitting Procedure for Simulating Hydro- and Thermal-Peaking Waves in Open-Channel Flows

Hydro- and thermal-peaking waves, generated by hydroelectric power generation, have a strong impact on the ecological integrity of aquatic ecosystems. In order to reduce such effects, mitigation procedure must be studied and implemented. To this end a one-dimensional model which solves the coupling of hydrodynamics with heat transport is developed. The solution is obtained advancing simultaneously the hydrodynamic and thermal module with the same accuracy. For the numerical solution of the governing advection-reaction/diffusion problem a splitting procedure is adopted: the advection-reaction part is solved by means of the weight average flux (WAF) finite volume explicit method, while the diffusion part is solved using a nonlinear version of the implicit Crank-Nicolson method. The WAF method is extended to second-order in the presence of reaction terms. Numerical results are presented for different test examples, which demonstrate the accuracy and robustness of the scheme and its applicability in predicting temperature transport by shallow water flows. Application to the Adige River (Northern Italy) of this framework proves that the model is an effective tool for designing hydro- and thermal-peaking waves mitigation procedures.

An unstructured-grid finite-volume surface wave model (FVCOM-SWAVE): Implementation, validations and applications

The structured-grid surface wave model SWAN (Simulating Waves Nearshore) has been converted into an unstructured-grid finite-volume version (hereafter referred to as FVCOM-SWAVE) for use in coastal ocean regions with complex irregular geometry. The implementation is made using the Flux-Corrected Transport (FCT) algorithm in frequency space, the implicit Crank–Nicolson method in directional space and options of explicit or implicit second-order upwind finite-volume schemes in geographic space. FVCOM-SWAVE is validated using four idealized benchmark test problems with emphasis on numerical dispersion, wave-current interactions, wave propagation over a varying-bathymetry shallow water region, and the basic wave grow curves. Results demonstrate that in the rectangular geometric domain, the second-order finite-volume method used in FVCOM-SWAVE has the same accuracy as the third-order finite-difference method used in SWAN. FVCOM-SWAVE was then applied to simulate wind-induced surface waves on the US northeast shelf with a central focus in the Gulf of Maine and New England Shelf. Through improved geometric fitting of the complex irregular coastline, FVCOM-SWAVE was able to robustly capture the spatial and temporal variation of surface waves in both deep and shallow regions along the US northeast coast.

A time-accurate variable property algorithm for calculating flows with large temperature variations

A variable property algorithm is developed for time-dependent resolution of flows with large temperature differences. The non-linear coupled set of equations with property variations is resolved by an implicit iterative procedure at each time step. The momentum and energy equations are integrated in time using an implicit Crank–Nicolson method, and a Helmholtz equation for pressure is solved at each inner iteration. Local density changes are coupled to both changes in local temperature and pressure, whereas other properties are only coupled to changes in temperature. No low Mach number assumption is used in the formulation, and the pressure is calculated directly through the Helmholtz equation. Results are shown to compare well with benchmark calculations and analytical solutions of convection in a differentially heated cavity with very large temperature differences. Unsteady Poiseuille–Bénard flow is used for validation of the time-dependent aspects of the algorithm, which is shown to accurately predict the time-dependent buoyancy driven longitudinal vortices.

Numerical solution of a complete surface energy balance model for simulation of heat fluxes and surface temperature under bare soil environment

Surface energy balance model is an essential approach for heat flux and evaporation estimation in applied meteorology and hydrology. Due to the complexity of soil–air interface system, the model has been simplified for different purposes in many researches. A complete model with full description of its complex factor relationships and its numerical solution has not been yet implemented in practical use. This paper presents a complete surface energy balance model with its inner relations cited from different researches. The model couples soil temperature change simultaneously with soil moisture movement, which makes the solution of the model uneasy. A detailed methodology of numerical approximation to the complete model is presented in the study for practical use. Soil heat and latent heat fluxes in the model are determined according to both soil temperature change and soil moisture movement, which are described as two differential equations. Crank–Nicolson implicit method is used to expand the differential equations into two sets of simultaneous linear equations, which are then solved by applying Gauss's elimination method. Latent heat flux is determined at the balance when evaporation from the surface is equal to the soil water loss. And surface temperature is estimated as the heat fluxes of the surface reaching the status of balance. The iterative computation of Newton–Raphson method is used to approximate latent heat flux and surface temperature from the balances. Based on this complexity of the model's relationships, a detailed computation procedure of the model is proposed. The methodology has been validated through application to south Israeli desert for heat flux and surface temperature estimation. A good matching of the simulated soil temperature to the measured one proves the validity of the model and method used for its numerical solution.

Deposition of Spherical Particles onto Cylindrical Solid Surfaces: I. Numerical Simulations

In this article, analytical and numerical solutions are given for the deposition of spherical particles onto cylindrical solid surfaces. If only the retarded van der Waals (vdW) force and the electrical double-layer (EDL) force between the spherical particle and the cylindrical surface are considered, such a deposition process is governed by a one-dimensional (1D) mass transfer equation for which both analytical and numerical solutions are obtained. A parametric study was conducted to examine the effects of these two colloidal forces on the deposition process. The numerical results show that the attractive vdW force, as represented by the dimensionless adhesion number Ad, is a necessary condition for deposition to occur. However, it affects only the drop number concentration distribution and, to a limited extent, the mass transfer rate. In general, the 1D deposition process is dominated by the EDL force, as represented by the dimensionless EDL parameter Dl and the reduced radius τ=κa of the spherical drop. It is noted that τ plays a key role in determining the effect of the EDL force on the mass transfer rate since an appreciable EDL force exists only within small separation distances (≤3τ−1). In addition to the two colloidal forces, the deposition process was also investigated in the presence of gravity and with a vertically applied electrical force. With these two external forces, the deposition becomes a two-dimensional (2D) boundary value problem for which a numerical solution was achieved using the implicit Crank–Nicolson method. Their effects on the 2D deposition process were studied.

An inverse problem of finding a source parameter in a semilinear parabolic equation

An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.

Laminar heat transfer for thermally developing flow of a herschel-bulkley fluid in a square duct

Laminar heat transfer of a Herschel-Bulkley fluid in the entrance region of a square duct assuming fully developed velocity profile is investigated numerically. The non linear momentum equation in fully developed region is solved iteratively using a finite difference method to obtain the velocity distribution. Using the velocity profile, the energy equation with viscous dissipation effects is solved by using an implicit Crank-Nicolson method to obtain the temperature distributions. Computations have been obtained for two cases T and H2 thermal boundary conditions. The product of the friction factor and Reynolds number, bulk mean temperatures and local Nusselt numbers for the two cases have been represented for various value of the model parameters and the Brinkman number. The present results have been compared with the known solutions for Newtonian and power-law fluids and are found to be in good agreement.

Evaluation of Hopscotch Method for Transient Ground-Water Flow

The hopscotch finite-difference technique is shown to be a fast and accurate way to simulate transient, saturated, ground-water flow in relatively typical but heterogeneous 2D and 3D domains. The odd-even hopscotch (OEH) and line hopscotch methods are reviewed, and their implementation for saturated ground-water flow is presented. The OEH scheme, which is a second-order accurate explicit process, is efficient, requiring only six floating point operations per mesh node and time step, and is unconditionally stable (for saturated ground-water flow). Numerical experiments on typical 2D meshes (2,500 nodes) with synthetic, randomly heterogeneous hydraulic conductivity, suggest that the OEH process is approximately 1.5 times faster than the alternating direction implicit method and 3–4 times faster than the Crank-Nicolson implicit method using preconditioned conjugate gradient iteration. Similar experiments on medium-sized 3D meshes (87,500 nodes) suggest that the OEH process is between 7 and 10 times faster than the Crank-Nicolson preconditioned conjugate gradient method. Although the numerical results presented illustrate only typical test problem performance, they nevertheless clearly indicate promise for using OEH to simulate transient ground-water flow in 2D and, especially, 3D heterogeneous domains requiring fine spatial meshes.

Numerical analysis of sorption and diffusion in soil micropores, macropores, and organic matter

Sorption and diffusion of volatile organic compounds (VOCs) in soil are important processes in subsurface contamination, but are complicated by the heterogeneous nature of soil. The soil pore-size distribution is herein approximated as a biporous geometry of micropores and macropores. The microspheres embedded in macrospheres contain micropores. Voids between microspheres constitute the macropores. Soil organic matter is assumed present on the surface of microspheres. We develop numerical solutions for the transient diffusion processes with linear partitioning in soil organic matter and nonlinear adsorption in the biporous soil particles. The three coupled nonlinear partial differential equations describe the diffusion in micropores and organic matter in series with diffusion in macropores. The computation uses an implicit Crank-Nicolson method with a predictor-corrector scheme for fast and efficient calculations. The solutions are applied to toluene and water sorption-desorption experimental data obtained by gravimetric uptake measurements. The procedure allows the estimation of key parameters, including micropore diffusivities.

ACTIVE COOLING AND THERMAL STRESS REDUCTION IN A POROELASTIC HOLLOW SPHERE

Using the thermoporoelastic theory previously proposed, thermal stresses are analyzed that are induced in a fluid-saturated poroelastic hollow sphere, whose inner surface is exposed to burning gas and the wall of which is cooled by the fluid injection from its outer surface. Two loading histories are considered: (i) the sphere is subjected to a sudden rise in temperature of the gas and simultaneously pressurized at its outer surface to cool the wall; and (ii) steady-state cooling is abruptly disturbed by a sudden loss of pressurization and, after a while, is recovered. The main focus is on the effect of heat advection due to active fluid injection on the reduction of temperature and thermal stresses. Since the formulated problem is axisymmetric, the displacement field is decoupled from the temperature and pore pressure fields, which are still coupled to each other. The coupled nonlinear diffusion equations are solved by the Crank-Nicolson implicit method. It has been concluded that the active fluid pressurization and injection are effective in suppressing the maximum thermal hoop stress at the outer surface without the occurrence of the excess compressional stress at the inner surface.

THERMAL STRESSES IN A FLUID-SATURATED POROELASTIC HOLLOW SPHERE

By using the thermoporoelastic theory proposed previously, thermal stresses are analyzed that are induced in a fluid-saturated porous hollow sphere subjected to a sudden rise in temperature and pressure on its inner wall. Since the problem formulated is spherically symmetric, the displacement field is decoupled from the temperature and pore fluid pressure fields, which are still coupled with each other. Coupled diffusion equations for heat and fluid flows are solved by the Crank-Nicolson implicit method because they involve nonlinear and integral terms. The attention is focused on the effect of heat transfer by fluid flow through pores on the temperature and thermal stress distributions. This effect is very marked for the case where the fluid diffusivity is much larger than the thermal one. This suggests a possible control of thermal stresses by active fluid injection.

The use of alternating direction implicit methods to model diffusion phenomena in elliptic products

Three alternating direction implicit (ADI) methods plus the Crank-Nicolson implicit method and an 8-noded isoparametric finite element method (both included for comparison purposes), are considered to solve numerically transient thermal or mass diffusion problems in elliptic shaped bodies. Moreover, three domain discretization methodologies are proposed: polar cylindrical, unequally and equally spaced orthogonal curvilinear coordinates. In order to analyse indirectly the effectiveness of the methods in different discretizing techniques, all of them are applied to water diffusion during rice grain soaking, and the computed solutions are compared with experimental ones. The influence of various time and space discretizations upon the results is studied, and numerical stability constraints of some of the methods used, as well as performance orientations on personal computer applications are given.

Active Cooling and Thermal Stress Reduction by Use of Porous Materials. Hollow Cylinder Model.

By using the thermo-poro-elastic theory proposed previously by the author, thermal stresses are analyzed which are induced in a fluid-saturated porous hollow cylinder whose inner surface is heated by burning gas, the wall of which is cooled by the fluid injection from its ourter surface. Two stories of loading conditions are considered : (A) the cylinder is subjected to a sudden rise in temperature of the gas and simultaneously pressurized at its outer surface to cool the wall, and (B) steady state cooling is abruptly disturbed by a sudden loss of pressurization, and after a while it is recovered. Main attention is placed on the effect of heat advection due to active fluid injection on the reduction of temperature and thermal stresses. Since the formulated problem is axisymmetric, the displacement field is decoupled from the temperature and pore-pressure fields, which are still coupled to each other. The coupled nonlinear diffusion equations are solved by the Crank-Nicolson implicit method.

Thermal Stresses of a Fluid-Saturated Poro-Elastic Hollow Cylinder.

By using the thermo-poro-elastic theory proposed by the author previously, thermal stresses are analyzed which are induced in a fluid-saturated porous hollow cylinder subjected to a sudden rise in temperature and pressure on its inner wall. Since the problem formulated is axisymmetric, the displacement field is decoupled from the temperature and pore-pressure fields, which are still coupled with each other. Coupled diffusion equations for heat and fluid flows are solved by the Crank-Nicolson implicit method, because they involve nonlinear and integral terms. The attention is focused on the effect of heat transportation by fluid flow through pores on the temperature and thermal stress distributions. This effect is very marked for the case where fluid diffusivity is much lager than the thermal one, as suggests the possible control of thermal stresses by active fluid injection.

Some numerical experiments on fisher equation

Fisher's equation which represents a reactive-diffusive system and has an exact traveling wave solution is used to compare several finite-differennce schemes. Iterative implicit, quasi-linear implicit, predictor-corrector explicit, and two forms of operator-splitting formulations have been examined. A stiffness parameter is introduced in the reaction term to vary the stiffness of the Fisher's equation. The numerical methods are compared on the basis of effectiveness curves. The results indicate that the time-linearization method is superior to the quasi-linearization method, that the Crank-Nicolson formulation is better than the standard implicit formulation, that the predictor-corrector explicit method is more effective than the Crank-Nicolson implicit method, that the two-step operator-splitting method is superior to the one-step operator-splitting method, and that the two-step operator-splitting with a fourth-order accurate solver for the reaction term is the most effective method.

Numerical simulation of convective diffusion at a rectangular channel flow electrode

The magnitude of the current is numerically computed for a mass transfer limited electrochemical reaction at a plane electrode in a rectangular channel under conditions of steady state, fully developed laminar flow. The procedure is based on the backward implicit finite difference numerical method, applied to solve the differential equation governing convective diffusion. The, convergence and error limits inherently present in the procedure are evaluated over a wide range of conditions, and compared with results obtained for the explicit finite difference method and the implicit Crank-Nicolson method. The backward implicit method is highly advantageous, for solving problems of the type investigated. Very good agreement is obtained with other experimental and semi-empirical studies over a wide range of operational parameters of the channel electrode.

Numerical simulation of convective diffusion at a rectangular channel flow electrode

The magnitude of the current is numerically computed for a mass transfer limited electrochemical reaction at a plane electrode in a rectangular channel under conditions of steady state, fully developed laminar flow. The procedure is based on the backward implicit finite difference numerical method, applied to solve the differential equation governing convective diffusion. The, convergence and error limits inherently present in the procedure are evaluated over a wide range of conditions, and compared with results obtained for the explicit finite difference method and the implicit Crank-Nicolson method. The backward implicit method is highly advantageous, for solving problems of the type investigated. Very good agreement is obtained with other experimental and semi-empirical studies over a wide range of operational parameters of the channel electrode.

Radial temperature differences during the melt spinning of fibers

In this investigation, a numerical model was developed to predict the temperature distribution in a fiber during melt spinning. This model uses the implicit Crank–Nicolson method to solve the governing differential equation for the problem. The model was applied to a series of numerical experiments on a liquid crystalline fiber which is melt-spun. These simulations used typical sets of operating conditions to determine the effect of various operating parameters on the predicted radius profile, spinline tension, and temperature distribution. The effects of spinneret capillary diameter, mass flow rate, ambient air temperature, spinning temperature, and elongational viscosity were investigated. The results of the various runs showed that ambient air temperature and mass flow rate had a significant effect on the predicted radius profile, spinline tension, and temperature distribution. The spinning temperature was an important parameter, but its only significant effect was on the spinline tension. Spinneret capillary diameter and elongational viscosity had little effect on the predicted results.