Crank-Nicholson Scheme Research Articles

Simulations of X-ray focusing by zone plates in rotationally symmetric optical field utilizing the matrix-free Finite Difference Beam Propagation Method

We present the use of a finite difference method based on Crank-Nicholson scheme and recurrence scheme for computationally efficient simulation of the X-ray propagation through a zone plate. By introducing boundary and central conditions and by avoiding large matrix operations, the method achieves considerable speed, little memory occupation and low background noise. Accommodating refractive index profiles of arbitrary shape, it can be applied to assist optimizing X-ray zone plates and understanding focusing mechanism.

Energetic particles transport in constants of motion space due to collisions in tokamak plasmas

The spatio-temporal evolution of the energetic particles in the transport time scale in tokamak plasmas is a key issue of the plasmas confinement, especially in burning plasmas. In order to include sources and sinks and collisional slowing down processes, a new solver, ATEP-3D was implemented to simulate the evolution of the energetic particle (EP) distribution in the three-dimensional constants of motion (CoM) space. The Fokker–Planck collision operator represented in the CoM space is derived and numerically calculated. The collision coefficients are averaged over the unperturbed orbits to capture the fundamental properties of EPs. ATEP-3D is fully embedded in ITER IMAS framework and combined with the LIGKA/HAGIS codes. The finite volume method and the implicit Crank-Nicholson scheme are adopted due to their optimal numerical properties for transport time scale studies. ATEP-3D allows the analysis of the particle and power balance with the source and sink during the transport process to evaluate the EP confinement properties.

Nonlinear system dynamics of calcium and nitric oxide due to cell memory and superdiffusion in neurons

The integer-order interdependent calcium ([Ca2+]) and nitric oxide (NO) systems are unable to shed light on the influences of the superdiffusion and memory in triggering Brownian motion (BM) in neurons. Therefore, a mathematical model is constructed for the fractional-order nonlinear spatiotemporal systems of [Ca2+] and NO incorporating reaction-diffusion equations in neurons. The two-way feedback process between [Ca2+] and NO systems through calcium feedback on NO production and NO feedback on calcium through cyclic guanosine monophosphate (cGMP) with plasmalemmal [Ca2+]-ATPase (PMCA) was incorporated in the model. The Crank–Nicholson scheme (CNS) with Grunwald approximation along spatial derivatives and L1 scheme along temporal derivatives with Gauss–Seidel (GS) iterations were employed. The numerical outcomes were analyzed to get insights into superdiffusion, buffer, and memory exhibiting BM of [Ca2+] and NO systems. The conditions, events and mechanisms leading to dysfunctions in calcium and NO systems and causing different diseases like Parkinson’s were explored in neurons.

Modelling Cross Talk in the Spatiotemporal System Dynamics of Calcium, IP3 and Nitric Oxide in Neuron Cells.

The bioenergetic system of calcium ([Ca2+]), inositol 1, 4, 5-trisphophate (IP3) and nitric oxide (NO) regulate the diverse mechanisms in neurons. The dysregulation in any or all of the calcium, IP3 and nitric oxide dynamics may cause neurotoxicity and cell death. Few studies are noted in the literature on the interactions of two systems like [Ca2+] with IP3 and [Ca2+] with nitric oxide in neuron cells, which gives limited insights into regulatory and dysregulatory processes in neuron cells. But, no study is available on the cross talk in dynamics of three systems [Ca2+], IP3 and NO in neurons. Thus, the cross talk in the system dynamics of [Ca2+], IP3 and NO regulation processes in neurons have been studied using mathematical model. The two-way feedback process between [Ca2+] and IP3 and two-way feedback process between [Ca2+] and NO through cyclic guanosine monophosphate (cGMP) with plasmalemmal [Ca2+]-ATPase (PMCA) have been incorporated in the proposed model. This coupling handles the indirect two-way feedback process between IP3 and nitric oxide in neuronal cells automatically. The numerical outcomes were acquired by employing the finite element method (FEM) with the Crank-Nicholson scheme (CNS). The present model incorporating the sodium-calcium exchanger (NCX) and voltage-gated calcium channel (VGCC) provides novel insights into the various regulatory and dysregulatory processes due to buffer, IP3-receptor, ryanodine receptor, cGMP kinetics through PMCA channel, etc. and their impacts on the interactive spatiotemporal system dynamics of [Ca2+], IP3 and NO in neurons. It is concluded that the behavior of different crucial mechanisms is quite different for interactions of two systems of [Ca2+] and NO and the interactions of three systems of [Ca2+], IP3 and nitric oxide in neuronal cell due to mutual regulatory adjustments. The association of several neurological disorders with the alterations in calcium, IP3 and NO has been explored in neurons.

Unsteady MHD Flow of Casson Hybrid Nanofluid Over an Infinite Vertical Flat Plate: Crank-Nicholson Scheme and Statistical Approach

Unsteady MHD Flow of Casson Hybrid Nanofluid Over an Infinite Vertical Flat Plate: Crank-Nicholson Scheme and Statistical Approach

Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations

PurposeThe purpose of this work is to introduce an efficient, global second-order accurate and parameter-uniform numerical approximation for singularly perturbed parabolic differential-difference equations having a large lag in time.Design/methodology/approachThe small delay and advance terms in spatial direction are handled with Taylor's series approximation. The Crank–Nicholson scheme on a uniform mesh is applied in the temporal direction. The derivative terms in space are treated with a hybrid scheme comprising the midpoint upwind and the central difference scheme at appropriate domains, on two layer-resolving meshes namely, the Shishkin mesh and the Bakhvalov–Shishkin mesh. The computational effectiveness of the scheme is enhanced by the use of the Thomas algorithm which takes less computational time compared to the usual Gauss elimination.FindingsThe proposed scheme is proved to be second-order accurate in time and to be almost second-order (up to a logarithmic factor) uniformly convergent in space, using the Shishkin mesh. Again, by the use of the Bakhvalov–Shishkin mesh, the presence of a logarithmic effect in the spatial-order accuracy is prevented. The detailed analysis of the convergence of the fully discrete scheme is thoroughly discussed.Research limitations/implicationsThe use of second-order approximations in both space and time directions makes the complete finite difference scheme a robust approximation for the considered class of model problems.Originality/valueTo validate the theoretical findings, numerical simulations on two different examples are provided. The advantage of using the proposed scheme over some existing schemes in the literature is proved by the comparison of the corresponding maximum absolute errors and rates of convergence.

A second order fractional step hybrid numerical algorithm for time delayed singularly perturbed 2D convection-diffusion problems

The objective of this work is to provide an efficient and higher-order numerical scheme for the solution of a singularly perturbed 2D time-delayed parabolic convection-diffusion problem. Each time step is split into two partial time steps by using the Peaceman-Rachford splitting algorithm. The splitting is done over the idea of the Crank-Nicholson scheme. The hybrid scheme is a combination of the central difference scheme in the layer region and the mid-point upwind scheme in the outer region defined on a Shishkin mesh for the discretization in the spatial direction. This makes the order of convergence to two (up to a logarithmic factor) in space. Again, the presence of the logarithmic effect is rectified by the use of the Bakhvalov-Shishkin mesh. The proposed method is proved to be parameter uniform and also it attains second-order spatial accuracy in the discrete supremum norm. Numerical tests validate the efficacy of the proposed scheme.

Hermite Interpolation Approach to High-Order Approximation of Heat Equations


 
 
 It is usually desirable to approximate the solution of mathemati- cal problems with high-order of accuracy and preferably using com- pact stencils. This work presents an approach for deriving high-order compact discretization of heat equation with source term. The key contribution of this work is the use of Hermite polynomials to reduce second order spatial derivatives to lower order derivatives. This does not involve the use of the given equation, so it is universal. Then, Tay- lor expansion is used to obtain a compact scheme for first derivatives. This leads to a fourth-order approximation in space. Crank-Nicholson scheme is then applied to derive a fully discrete scheme. The result- ing scheme coincides with the fourth-order compact scheme, but our derivation follows a different philosophy which can be adapted for other equations and higher order accuracy. Two numerical experiments are provided to verify the fourth-order accuracy of the approach.
 
 

A quasi-implicit and coupled multiscale scheme for simulating combustion of pellets of core-shell structured intermetallic particles

A comprehensive theoretical and computational framework is developed to simulate combustion and flame propagation in a packed pellet of core-shell structured intermetallic energetic particles. A coupled multiscale approach is adopted, in which the atomic diffusion process in micron-scale particles is strongly linked to the macro-scale energy transport in the pellet. Species transport equations are solved for the core-shell structured particle, with the diffusivity treated as a temperature-dependent parameter. The energy equation is solved for the pellet, with the source term governed by the evolution of species concentrations in the core-shell structured particle. A quasi-implicit numerical scheme is developed to ensure stability and a generalized Crank-Nicholson scheme is adopted to tune the degree of implicitness and control the order of temporal accuracy. Further, the temperature dependence of properties and phase change of materials are considered. The theoretical and computational framework is applied to simulate combustion of pellets with Ni/Al core-shell particles of diameters in the range of 10–100 µm. An attempt is made to simulate experiments as closely as possible and the predictions are compared with the available experimental data. Suitable diffusivity model parameters that best describes the experimental data are identified. Parametric studies are conducted to study the effects of particle size, interstitial gas, and pressure on the combustion velocity. Results provide novel insights on the combustion process and directions for future studies.

Longitudinal waves in the walls of an annular channel filled with liquid and made of a material with fractional nonlinearity

Purpose of this paper is to study the evolution of longitudinal strain waves in the walls of an annular channel filled with a viscous incompressible fluid. The walls of the channel were represented as coaxial shells with fractional physical nonlinearity. The viscosity of the fluid and its influence on the wave process was taken into account within the study. Metods. The system of two evolutionary equations, which are generalized Schamel equations, was obtained by the two-scale asymptotic expansion method. The fractional nonlinearity of the channel wall material leads to the necessity to use a computational experiment to study the wave dynamics in them. The computational experiment was conducted based on obtaining new difference schemes for the governing equations. These schemes are analogous to the Crank–Nicholson scheme for modeling heat propagation. Results. Numerical simulation showed that over time, the velocity and amplitude of the deformation waves remain unchanged, and the wave propagation direction concurs with the positive direction of the longitudinal axis. The latter specifies that the velocity of the waves is supersonic. For a particular case, the coincidence of the computational experiment with the exact solution is shown. This substantiates the adequacy of the proposed difference scheme for the generalized Schamel equations. In addition, it was shown that solitary deformation waves in the channel walls are solitons.

Hall and ion slip effects on hybrid nanoparticles (Au-GO/blood) flow through a catheterized stenosed artery with thrombosis

Present paper deals with the effect of Hall and ion slips on MHD blood flow through a catheterized multi-stenosis artery with thrombosis. Blood is considered a base fluid, and the suspension of Au and GO nanoparticles are used to form the hybrid nano-blood suspension. The hemodynamic mathematical model is presented to mimic the blood flow inside the artery subjected to Joule heating, thermal radiation, Hall and ion slip effects. The effect of the shape of nanoparticles and the variable viscosity depending upon hematocrit is taken into account. The mathematical model is presented in a curvilinear coordinate system, and a mild stenosis assumption is used to get the closed-form solutions. The reduced governing equations are solved using the Crank Nicholson scheme with “MATLAB.” The effect of various pertinent parameters on flow patterns are illustrated graphically. It is observed that increasing the Hall and ion slips parameter causes the increment in the fluid velocity due to an enhancement in the cyclotron frequency of the particles. The hybrid nanoparticles Au-GO/blood has a higher temperature profile than pure blood and unitary nanoparticles, as the thermal conductivity increases with an increase in nanoparticle concentration. Scientists and clinical researchers may find the present study useful in understanding the various illness’s conditions, including cancers, infections, and the removal of blood clots. Furthermore, current research may be helpful in the biomedical area, such as magnetic resonance angiography (MRA), which creates an image of an artery to identify any abnormalities in the artery. The current findings are consistent with recent findings in earlier blood flow research studies.

Solution to the problem of thermal conductivity in a two-dimensional formulation on a graphics processing unit using parallel computing

Subject: technology and algorithms of parallel programming.
 Objective: to compare the execution speed of a sequential algorithm on a central processor with a parallel algorithm on a graphics processor when solving a two-dimensional thermal conductivity problem.
 Methods: the Crank Nicholson method modified by the author for solving the problem of two-dimensional thermal conductivity on a graphics processor.
 Research results: 1. The solution of the two-dimensional thermal conductivity problem according to the Crank Nicholson scheme is not absolutely parallel and the maximum possible acceleration is not achieved 2. With a matrix dimension of 16 by 16 and single precision, the execution time on the GPU turned out to be 1.32 1.72 times faster than on the CPU. With a 32 by 32 matrix dimension, the execution time on the GPU turned out to be 3.66 6.07 times faster than on the CPU. 3. When calculating with double precision, the greatest acceleration is observed at 71.89 with 10 iterations of calculation, if there are more than 104 iterations, then the acceleration in calculations on a GPU with double precision approaches calculations with single precision.

Numerical investigation of the magnetic properties and behavior of electrically conducting fluids via the local weak form method

The magnetohydrodynamics (MHD) equation is one of the practical models that it has different applications in physics and engineering. It describes the magnetic properties and behavior of electrically conducting fluids. This paper investigates the MHD equation numerically. First, the time derivative is approximated by utilizing the Crank–Nicholson scheme. The convergence order and unconditional stability of the proposed time-discrete scheme are analyzed by the energy method. Then, the direct meshless local Petrov–Glerkin (DMLPG) method is applied for the spatial derivative to get the full-discrete scheme. The new numerical procedure is used to simulate this equation in a pipe with regular and irregular cross-sectional area. Numerical results show the efficiency of this method is well.

A discrete algorithm for general weakly hyperbolic systems

This paper studies the Cauchy problem for first order systems, 0.1 $$\begin{aligned} Lu \ =\ \partial _tu-\sum _{j=1}^dA_j(t,x)\partial _{x_j}u-B(t,x)u\ =\ f, \quad u(0,\cdot ) = g. \end{aligned}$$ Assume that for $$\xi \in \mathbb {R}^d,$$ $$\sum A_j\xi _j$$ has only real eigenvalues. For coefficients and Cauchy data sufficiently Gevrey regular the Cauchy problem has a unique sufficiently Gevrey regular solution. We prove stability and error estimates for the spectral Crank-Nicholson scheme. Approximate solutions can be computed with accuracy $$\epsilon $$ in $$L^\infty ( [0,T]\times \mathbb {R}^d)$$ with cost growing at most polynomially in $$\epsilon ^{-1}$$ . The proofs uses pseudodifferential symmetrizers.

Stability Analysis of the Crank-Nicolson Finite Element Method for the Navier-Stokes Equations Driven by Slip Boundary Conditions

This paper is devoted to the study of numerical approximation for a class of two-dimensional Navier-Stokes equations with slip boundary conditions of friction type. The objective is to establish the well-posedness and stability of the numerical scheme in L 2 -norm and H 1 -norm for all positive time using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining H 1 -estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs. We prove that the numerical scheme is stable in L 2 and H 1 -norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Crank Nicholson scheme to examine the fractional-order unsteady nanofluid flow of free convection of viscous fluids.

Fractional fluid models are usually difficult to solve analytically due to complicated mathematical calculations. This difficulty in considering fractional model further increases when one considers nth order chemical reaction. Therefore, in this work an incompressible nanofluid flow as well as the benefits of free convection across an isothermal vertical sheet is examined numerically. An nth order chemical reaction is considered in the chemical species model. The specified velocity (wall's) is time-based, and its motion is translational into mathematical form. The fractional differential equations are used to express the governing flow equations (FDEs). The non-dimensional controlling system is given appropriate transformations. A Crank Nicholson method is used to find solutions for temperature, solute concentration, and velocity. Variation in concentration, velocity, and temperature profiles is produced as a result of changes in discussed parameters for both Ag-based and Cu-based nanofluid values. Water is taken as base fluid. The fractional-order time evaluation has opened the new gateways to study the problem into a new direction and it also increased the choices due to the extended version. It records the hidden figures of the problem between the defined domain of the time evaluation. The suggested technique has good accuracy, dependability, effectiveness and it also cover the better physics of the problem specially with concepts of fractional calculus.

ЕФЕКТИВНIСТЬ БАГАТОПОТОЧНОГО РОЗПАРАЛЕЛЮВАННЯ ПРИ МОДЕЛЮВАННI ВОЛОГОПЕРЕНЕСЕННЯ НА ОСНОВI ДРОБОВО-ДИФЕРЕНЦIАЛЬНОЇ МОДЕЛI НА AMD EPYC

The paper studies the performance of multithreaded parallel implementation of a finite-difference solver for a two-dimensional space-fractional generalization of Richards equation. For numerical solution we used implicit Crank-Nicholson scheme with L1-approximation of Caputo fractional derivative and TFQMR linear systems’ solver. OpenMP implementation was tested on three CPUs — server Intel Xeon Bronze 3104 and AMD EPYC 7542 along with laptop AMD Ryzen 3 5300U. Testing results show that the proposed implementation can give close-to-linear acceleration when executing on up to 8 cores. On high-performance AMD EPYC maximal acceleration was achieved when 32-64 cores were used showing limited scalability of the algorithms on such a CPU.

Radiation and heat absorption effects on an unsteady MHD boundary layer flow along an accelerated infinite vertical plate with ramped plate temperature in the existence of slip condition

A numerical analysis of an unsteady MHD laminar boundary layer flow over an accelerated moving vertical plate along with ramped temperature was performed, in which chemical reaction, heat absorption, and radiation were considered. Slip circumstances have sparked much concern, because of its widespread application in chemical engineering and manufacturing industries. The equations of momentum, thermal, and as well as species concentration, were solved numerically by implementing the Crank–Nicholson scheme. There is a visual representation of changes in fluid velocity, temperature, and concentration, the numerical values of drag force, Nusselt, and Sherwood numbers are provided in tabular form for various values of field variables. Results show that slip condition has a considerable impact on a small intensity of λ, and that this effect is extra pronounced in the steady condition than in the ramped condition. Profiles of velocity and temperature are reducing in both cases for increased values of heat absorption and magnetic field parameters.

Investigation of heat transfer in metal nanofilms irradiated with ultrashort laser pulses: two-temperature model

A numerical study of heat transfer between an electron gas and a crystal lattice in a metal nanofilm under irradiation with an ultrashort laser pulse was carried out on the basis of a parabolic two-temperature model of thermal conductivity presented in a dimensionless form. For the numerical solution, the finite difference method was used using the explicit-implicit Crank-Nicholson scheme. As a result of the numerical study, it was found that with an increase in the thickness of the plate, the equilibrium temperature decreases, and the time for the onset of thermal equilibrium between the electrons and the crystal lattice increases.

Septic Hermite collocation method for the numerical solution of Benjamin–Bona–Mahony–Burgers equation

In this paper, the nonlinear Benjamin–Bona–Mahony–Burgers (BBMB) equation is solved using the septic Hermite collocation method (SHCM) for the space discretization and Crank–Nicholson scheme for the time discretization. The Von-Neumann stability analysis is used to show the unconditional stability of the proposed method. The error bounds for the septic Hermite interpolation polynomials are estimated using the Peano's theorem. The method is applied to one, two and three interaction solitary wave problems. The and error norms are calculated to demonstrate the efficiency and the robustness of the proposed technique.