Implicit Crank-Nicolson Method Research Articles

Optimal [formula omitted] error estimates of mass- and energy- conserved FE schemes for a nonlinear Schrödinger–type system

In this paper, we present an implicit Crank–Nicolson finite element (FE) scheme for solving a nonlinear Schrödinger–type system, which includes Schrödinger–Helmholz system and Schrödinger–Poisson system. In our numerical scheme, we employ an implicit Crank–Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal L2 error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.

Thermal and mass stratification effects on MHD nanofluid past an exponentially accelerated vertical plate through a porous medium with thermal radiation and heat source

The aim of this research is to investigate the combined effects of thermal and mass stratification on unsteady magnetohydrodynamic nanofluid moving through an exponentially accelerated vertical plate in a porous medium. The governing equations of the problem are solved numerically by using the implicit Crank–Nicolson method. The results of the nanofluid with two different stratification are compared to those obtained without any stratification. The velocity of the nanofluid decreases with both types of stratification, whereas temperature and concentration decrease with thermal stratification and mass stratification, respectively. A variety of parameters, including the volume fraction of nanoparticles, thermal radiation, heat source/sink, and chemical reaction, can be studied using graphs. The important results demonstrate that nanofluids are more thermally conductive than normal fluids. This research holds significant implications in various fields, including power generation, electronic component cooling, and vehicle construction.

THERMAL DIFFUSION AND RADIATION ABSORPTION IMPACTS ON CHEMICALLY REACTIVE AND RADIATIVE NANOFLUID FLOW PAST AN EXPONENTIALLY ACCELERATED VERTICAL POROUS PLATE

The role of thermal diffusion (Soret) and radiation-absorption influences on the unsteady MHD boundary layer flow for Silver (Ag) and Copper (Cu) water-based nanofluid over an accelerated exponentially vertical porous plate has been investigated. The varying temperature and concentration at the plate are also considered. The governing equations of the flow are numerically resolved to employ the finite-difference system of the implicit Crank-Nicolson method. Fluid velocity, f luid temperature, and species concentration were all plotted for meaningful physical parameters. For both nanofluids, non-dimensional metrics such as the friction factor, Nusselt number, and Sherwood number at the surface are tabulated. The data obtained from the literature provide strong validation for the findings. The f inding of the study is that both nanofluids’ velocities rise with increases in Soret and radiation absorption parameters. For increased values of Soret and radiation absorption parameters, the friction factor and Nusselt number increase. Sherwood numbers drop for the Soret parameter and rise for the radiation absorption.

Optimal Convergence of the Newton Iterative Crank–Nicolson Finite Element Method for the Nonlinear Schrödinger Equation

Abstract An error estimate is presented for the Newton iterative Crank–Nicolson finite element method for the nonlinear Schrödinger equation, fully discretized by quadrature, without restriction on the grid ratio between temporal step size and spatial mesh size. It is shown that the Newton iterative solution converges double exponentially with respect to the number of iterations to the solution of the implicit Crank–Nicolson method uniformly for all time levels, with optimal convergence in both space and time.

Linearly implicit variable step-size BDF schemes with Fourier pseudospectral approximation for incompressible Navier-Stokes equations

In this paper, linearly implicit backward differentiation formulas with variable step-sizes are proposed to solve numerically the two-dimensional incompressible Navier-Stokes equations (formulated in vorticity-stream function). With Fourier pseudospectral methods for spatial discretization, the diffusion term is discretized implicitly and the nonlinear convection term is treated by a combination of implicit and explicit discretizations. As a result, only linear solvers are needed at each time step to achieve the desired temporal accuracy. With the help of a priori assumption and aliasing error control techniques, the error estimates for one-step and two-step backward differentiation formulas are established in several norms under appropriate step-size constraints. Compared with the numerical results of implicit-explicit (the nonlinear convection term is treated explicitly) BDF2 method and fully implicit Crank-Nicolson method, it demonstrates that the proposed linearly implicit variable step-size BDF2 method is effective and robust.

Numerical investigation of chemical reaction, Soret and Dufour impacts on MHD free convective gyrating flow through a vertical porous channel

The extensive variety of issues include reactions kinetics, simulations and optimizations of different models of reactors including basic explorations of the processes of temperature, mass and momentum transfer that taken places with chemical reaction. Based on this criteria, it is discussed the impacts of the Soret and Dufour consequences on natural convective heat and mass transfer for the unsteady two dimensional boundary layer flow through a vertical channel/duct with the influence of viscous dissipation, invariable suction into the description. The governing dimensionless partial differential equations are solved digitally making use of the implicit Crank-Nicolson method. The velocity, temperature, and concentration dispensations are addressed computationally and demonstrated by the graphs. Numerical quantities of the skin-friction, Nusselt number and Sherwood number at the plate are found for a choice of assessments of substantial parameters and which are displayed by tabular manner. It is significant to notice that, Soret impacts emerge in suspended mixture of particles and fluids. These phenomenons are reasoned with the temperature gradients; hence, the movements of fluid molecules in the hottest zone and highest energy level in this region transfer the particles towards the coldest region. It is also most useful in chemical engineering.

Modeling and Computing of Fractional Convection Equation

In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in (0, 1). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in (0, 1) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank–Nicolson method and the explicit Lax–Wendroff method in time, and the second-order numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efficiency of the derived methods are verified by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.

Exponential B-spline collocation solutions to the Gardner equation

ABSTRACT Exponential B-splines are used to set up a collocation method for solving the Gardner equation. The space reduction of the Gardner equation is carried out to be able to obtain an exponential B-spline approximation for the collocation method. Thus, a coupled system is integrated using the Crank–Nicolson implicit method in time together with the first-order linearization method and then the collocation method is applied to have a linear algebraic system. This system is shown to be stable by using the Von Neumann analysis. The discrete maximum errors are found fairly small and relative changes of the conservation laws remain constant during simulations for the text problems.

Nonlinear Partial Dierential Equations Model Related to Oxidation Pond Treatment System: A Case Study of mPHO at Taman Timor Oxidation Pond, Johor Bahru

This study presents a mathematical model examining wastewater pollutant removalthrough an oxidation pond treatment system. This model was developed to describethe reaction between microbe-based product mPHO (comprising Phototrophic bac-teria (PSB)), dissolved oxygen (DO) and pollutant namely chemical oxygen demand(COD). It consists of coupled advection-diusion-reaction equations for the microor-ganism (PSB), DO and pollutant (COD) concentrations, respectively. The couplingof these equations occurred due to the reactions between PSB, DO and COD to pro-duce harmless compounds. Since the model is nonlinear partial dierential equations(PDEs), coupled, and dynamic, computational algorithm with a specic numericalmethod, which is implicit Crank-Nicolson method, was employed to simulate the dy-namical behaviour of the system. Furthermore, numerical results revealed that theproposed model demonstrated high accuracy when compared to the experimental data.Keywords Oxidation pond; nonlinear PDEs; PSB; implicit Crank-Nicolson.

Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes–Darcy problem

We propose and study a combination of two second-order implicit–explicit (IMEX) methods for the coupled Stokes–Darcy system that governs flows in karst aquifers. The first is a second-order explicit two-step MacCormack scheme and the second is a second-order implicit Crank–Nicolson method. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. This combination so called the MacCormack rapid solver method is very efficient (faster, at least of second order accuracy in time and space) and can be easily implemented using legacy codes. Under time step limitation of the form Δt≤Ch (where h,Δt are mesh size and time step, respectively, and C is a physical parameter) we prove both long time stability and the rate of convergence of the method. Some numerical experiments are presented and discussed.

FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this paper, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different types of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank–Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of 240Pu and 256Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. We emphasize that in a realistic calculation of fission mass distributions of 240Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents). Program summaryProgram Title: FELIX-2.0Program Files doi:http://dx.doi.org/10.17632/t8b4h9g88r.1Licensing provisions: GPLv3Programming language: C++Journal reference of previous version: Computer Physics Communications 200, 350–363 (2016)Does the new version supersede the previous version?: YesReasons for the new version: FELIX-2.0 extends the physics capabilities of the previous version, since it includes the option to have a metric term in the collective Schrödinger equation. The computational efficiency of the code is considerably increased thanks to the implementation of several new numerical methods. This version also provides a more flexible and robust workflow to perform calculation of fission fragment distributions more efficiently.Summary of revisions: The new version includes the ability to use a metric term in the TDGCM+GOA equation. Numerical integration methods (both spatial and time integration) have been updated to more efficient schemes: matrix elements are now computed with Gauss–Legendre or Gauss–Legendre–Lobatto quadratures, the time propagation of the wave packet is computed with the Krylov approximation of the propagator instead of the previous Crank–Nicolson scheme. A new type of finite element based on n-dimensional orthotope is implemented, which enables a spectral element scheme for spatial discretization. Finally, the workflow and inputs/outputs of the code were entirely redesigned to provide more flexibility and be more user-friendly.Nature of problem: The Gaussian overlap approximation to the time-dependent generator coordinate method [1,2] yields a local, time-dependent Schrödinger equation in a small, multi-dimensional collective space. Its solution provides the time-evolution of the collective probability amplitude. For applications to nuclear fission, scission configurations are defined by a hyper-surface in this collective space referred to as the frontier. Distributions of nuclear observables such as charge or mass distributions of the fission fragments are extracted from the probability for the system to pass through any given section of the frontier. This probability is computed by first solving the evolution equation up to times greater than 10−20s and then integrating the flux of probability through the frontier over the entire timerange.Solution method: FELIX-2.0 solves the time-dependent Schrödinger equation by first discretizing the N-dimensional collective space with the continuous Galerkin Finite Element Method. This produces a large set of coupled, time-dependent Schrödinger equations characterized by the sparse overlap and Hamiltonian matrices. The solution is evolved in small time steps by applying an explicit and unitary propagator built as a Krylov approximation [3] of the exponential of the Hamiltonian.Additional comments including Restrictions and Unusual features: Although the implementation of the program gives it the ability to solve the TDGCM+GOA equation in a generic N-dimensional collective space, it has only been tested on 1-, 2- and 3-dimensional meshes.[1] J.J. Griffin, J. A. Wheeler, Collective Motions in Nuclei by the Method of Generator Coordinates, Phys. Rev. C 108, 311-327 (1975)[2] P. G. Reinhard, K. Goeke, The Generator-Coordinate Method and Quantized Collective Motion in Nuclear Systems, Rep. Prog. Phys. 50, 1-64 (1987)[3] Y. Saad, Iterative Methods for Sparse Linear Systems: Second Edition, SIAM, 2003

Effect of Thermal Conductivity on Magnetohydrodynamic Heat and Mass Transfer in Porous Medium Saturated with Kuvshinshiki Fluid

The effects of Kuvshinshiki fluid on Magnetohydrodynamic (MHD) heat and mass transfer flow over a vertical porous plate with chemical reaction of nth order and thermal conductivity was carried out. The governing partial differential equations were solved numerically using implicit Crank-Nicolson method. A parametric study was performed to illustrate the impact of visco-elastic parameter, radiation parameter, thermal conductivity parameter, magnetic parameter, Prandtl number on the velocity,temperature and concentration profiles.The results were presented graphically with tabular presentations of the skin-friction,rate of heat and mass transfer which were all computed and discussed for different values of parameters of the problem. The numerical results revealed that the visco- elastic of Kuvshinshiki fluid type is growing as concentration profile increases, while the velocity and temperature profile falls ,then the radiation and thermal conductivity were growing as velocity and temperature increases. Also Sherwood number decreases as radiation increases but Sherwood number remains unchanged as thermal conductivity growing.

The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.

ATUS-PRO: A FEM-based solver for the time-dependent and stationary Gross–Pitaevskii equation

ATUS-PRO is a solver-package written in C++ designed for the calculation of numerical solutions of the stationary- and the time dependent Gross–Pitaevskii equation for local two-particle contact interaction utilising finite element methods. These are implemented by means of the deal.II library (Bangerth et al., 0000) [1], (Bangerth et al., 2007) [2]. The code can be used in order to perform simulations of Bose–Einstein condensates in gravito-optical surface traps, isotropic and full anisotropic harmonic traps, as well as for arbitrary trap geometries. A special feature of this package is the possibility to calculate non-ground state solutions (topological modes, excited states) (Marojević et al., 2013), (Yukalov et al., 1997, 2004) [3,4] for an arbitrarily high non-linearity term. The solver-package is designed to run on parallel distributed machines and can be applied to problems in one, two, or three spatial dimensions with axial symmetry or in Cartesian coordinates. The time dependent Gross–Pitaevskii equation is solved by means of the fully implicit Crank–Nicolson method, whereas stationary states are obtained with a modified version based on our own constrained Newton method (Marojević et al., 2013). The latter method enables to find the excited state solutions. Program summaryProgram title: ATUS-PROCatalogue identifier: AEZD_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEZD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 12757No. of bytes in distributed program, including test data, etc.: 412782Distribution format: tar.gzProgramming language: C++.Computer: PCs and distributed memory machines.Operating system: Linux, UNIX.RAM: Depending on the problem; megabytes to gigabytesSupplementary material: A file containing the expected results from the test run can be downloaded.Classification: 4.3, 4.12.External routines: MPI, GSL, LAPACK, P4EST, PETSC, deal.IINature of problem: Solving the Gross–Pitaevskii equation for Bose–Einstein condensates in external traps. Stationary solutions: computation of ground as well as excited states. Real time propagation: calculation of time dependent solutions.Solution method: The method of solving for stationary states is based on an enhanced version of the Newton algorithm developed in [1]. An implicit Crank–Nicolson scheme is used for real-time propagation. Both methods use adaptive finite element methods based on the library deal.II.Restrictions: The one-dimensional programs run only on single core.Additional comments: This package generates 8 executables, (i) breed_1, (ii) breed, (iii) breed_cs, (iv) rtprop_1, (v) rtprop, (vi) rtprop_cs, (vii) gen_params, (viii) gen_params_csRunning time: Depending on size of problem: from seconds (ground state calculations) to minutes (small no. of excited states, short timescale real-time propagation) up to several days (large no. of excited states and large scale real time propagation).

A note on the numerical approach for the reaction–diffusion problem to model the density of the tumor growth dynamics

In this article, we numerically solve an equation modeling the evolution of the density of glioma in the brain—the most malignant form of brain tumor quantified in terms of net rates of proliferation and invasion. We employ a non-linear heterogeneous diffusion logistic density model. This model assumes that glioma cell invasion throughout the brain is a reaction–diffusion process and that the coefficient of diffusion can vary according to the gray and white matter composition of the brain at that location. The analysis provided in this article demonstrates that using the correct finite difference scheme can overcome the stability issues caused by the discontinuities of the diffusion coefficient. We also observe that at the steady-state these discontinuities vanish. To visualize and investigate numerically the behavior of the evolution of tumor concentration of the glioma, we calculated and plotted the number of tumor cells, the average mean radial distance, and the speed of the tumor cells along with charting the effects of net dispersal rate and net proliferation rate terms versus time for different center position values of Gaussian initial profile for each zone (gray and white matter tissues). We have proposed two numerical methods, the implicit backward Euler and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method for solving the governing equations. These methods are compared in terms of their performance in varying time-step and mesh-discretization. The Crank–Nicolson implicit method is shown to be the better choice to solve the equation.

Studies on melt spinning of sea-island fibers. II. Dynamics of melt spinning of polypropylene/polystyrene blend fibers

A two-dimensional (2-D) model for melt spinning of iPP/aPS blend fibers is proposed based on two-phase models on density and crystallinity and log-additive rule on elongational viscosity. A computer program is developed based on a hybrid method of fourth-order Runge-Kutta method and implicit Crank-Nicolson method to solve the model equations to obtain the axial profiles of fiber diameter, velocity, gradient of velocity and crystallinity, and the 2-D profiles of temperature, elongational viscosity and elongational stress. The simulated fiber diameters are compared with the measured diameters to verify the certainness of the simulation. The simulated results show that polymer melt jets solidify at the positions of about 40 cm beneath the spinneret which is verified by on-line measurement of fiber diameter. And the radial gradient of temperature, elongational viscosity and elongational stress reaches to 104 to 105 °C/m, 105 to 106 Pa·s/m and 105 to 106 Pa/m, respectively, at the discussed take-up velocities.

On the Influence of Soret and Dufour Effects on MHD Free Convective Heat and Mass Transfer Flow over a Vertical Channel with Constant Suction and Viscous Dissipation

The present paper investigates the combined effects of Soret and Dufour on free convective heat and mass transfer on the unsteady one-dimensional boundary layer flow over a vertical channel in the presence of viscous dissipation and constant suction. The governing partial differential equations are solved numerically using the implicit Crank-Nicolson method. The velocity, temperature, and concentration distributions are discussed numerically and presented through graphs. Numerical values of the skin-friction coefficient, Nusselt number, and Sherwood number at the plate are discussed numerically for various values of physical parameters and are presented through tables. It has been observed that the velocity and temperature increase with the increase in the viscous dissipation parameter and Dufour number, while an increase in Soret number causes a reduction in temperature and a rise in the velocity and concentration.

A large-scale wave-current coupled module with wave diffraction effect on unstructured meshes

Based on the extended mild-slope equation, a large-scale wave module is developed. By combining the eikonal equation and the modified wave action equation, the wave model can account for diffraction in most situations such as in the lee of islands and breakwaters, and using unstructured meshes provides great flexibility for modelling the wave in the complex geomorphology of barriers and islands, also allowing for refinement of the grid resolution within computationally important domains. The numerical implementation of the module is based on the explicit second-order upwind finite-volume schemes in geographic space, the Flux-Corrected Transport (FCT) algorithm in frequency space and the implicit Crank-Nicolson method in directional space. The three-dimensional hydrodynamic module is then modified to couple with the wave model, where the wave readily provides the depth-dependent radiation stress and the wave-induced turbulence coefficient for the current fields, and the wave propagation takes into account the current-induced advection, refraction and diffraction of wave energy and the effect of water level. The applicability of the proposed model to calculate Snell’s Law, wave transformation over the breakwaters and the elliptic shoal, wave propagation over the rip current field and the undertow on a sloping beach is evaluated. Numerical results show that the present model makes better predictions of the near-shore wave propagation and complex three-dimensional (3D) near-shore circulation driven by the waves, considering analytical solutions and experimental values.

Pricing Asian call option with average strike using a non-uniform grid

The problem of pricing Asian call option with average strike on a non-uniform grid is examined. The backward partial differential equation (PDE) along with the final and boundary conditions for an Asian call option with average strike is presented. A Crank Nicolson Implicit Method based scheme on a non-uniform grid for both spatial and temporal direction is derived, making use of a transformation to generate the non-uniform grid. The scheme is implemented to obtain the price of the Asian option for various values of drift and volatility. The results show a close match with the values obtained using Monte Carlo simulation and numerical PDE methods on uniform grid. The values obtained using the non-uniform grid (as compared to uniform grid) were closer to the values from Monte-Carlo simulation for smaller volatilities, thereby indicating that non-uniform grid could lead to more accurate results in case of smaller volatilities.

Numerical Investigation of the Aerodynamic and Structural Characteristics of a Corrugated Airfoil

5). The majority of studies have been carried out with scaled models of dragon fly forewings from the Aeshna Cyanea in either wind tunnels or water channels. In this paper, we study the aerodynamics of a corrugated airfoil using computational fluid dynamics at a low Reynolds number of 1000. Structural analysis is also performed using the commercial software SolidWorks 2009. The flow field is described by solving the incompressible Navier-Stokes equations on an overlapping grid using pressure-Poisson method. The equations are discretized in space with second-order accurate central differences. Time integration is achieved through the secondorder Crank-Nicolson implicit method. The complex vortex structures that form in the corrugated airfoil valleys and around the corrugated airfoil is studied in detail. Comparisons are made with experimental measurement at different Reynolds numbers and also with simulations of a flat plate. Contrary to the studies at high Reynolds numbers, our study shows that at low Reynolds numbers the corrugation does not provide any aerodynamic benefit compared to a flat plate. Instead, the corrugated airfoil generates more drag than the flat plate. Structural analysis shows that the wing corrugation can increase the resistance to bending moments on the wing structure. A smoothed structure has to be three times thicker to provide the same resistance. We conclude the corrugated wing has the structural benefit to provide the same resistance to bending moments with a much reduced thickness and weight.