Crank-Nicolson Research Articles

Singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme

In this paper, we proposed an accurate ϵ-uniformly convergent numerical method to solve singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme. The time-fractional derivative is defined in the sense of Caputo with order η∈(0,1). The time-fractional derivative is discretized by employing the Crank–Nicolson method on a uniform mesh, and an exponential fitted operator scheme along with the standard upwind method is used to mesh-grid the space domain. The truncation error and uniform stability of the discretized problems are examined in order to prove the parameter uniform convergence of the proposed scheme. It is demonstrated that the scheme is ϵ-uniformly convergent of order O((Δt)2−η+Δx), where Δt and Δx represent the step sizes of the time and space domains, respectively. Two numerical examples are provided in order to assess the accuracy of the suggested scheme and validate the theoretical concepts discussed. To demonstrate the efficiency of the numerical scheme presented, comparisons have been made with the numerical solution obtained by the finite difference method that exists in the literature. Consequently, it is observed that the results obtained by the present scheme are more accurate and have a better convergence rate.

On the phase-field algorithm for distinguishing connected regions in digital model

In this paper, we propose a novel model for the discrimination of complex three-dimensional connected regions. The modified model is grounded on the Allen–Cahn equation. The modified equation not only maintains the original interface dynamics, but also avoids the unbounded diffusion behavior of the original Allen–Cahn equation. This advantage enables us to accurately populate and extract the complex connectivity region of the target part. The model is discretized employing a semi-implicit Crank–Nicolson scheme, ensuring second-order accuracy in both time and space. This paper provides a rigorous proof of the unconditional energy stability of our method, thereby affirming the numerical stability and the physical rationality of the solution. We validate the discriminative ability of the proposed model for 3D complex connected regions.

Numerical study of the multi-dimensional Galilei invariant fractional advection–diffusion equation using direct mesh-less local Petrov–Galerkin method

This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of O(τ2). In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

Isogeometric method for a nonlocal degenerate parabolic problem

In this article, we apply NURBS-based isogeometric method to solve a nonlocal degenerate parabolic problem. The isogeometric method has advantages over classical finite element method in terms of exact geometry representation and higher order smooth basis functions. We consider the NURBS-based semi-discretization of the problem. The second-order Crank–Nicolson method is applied for the complete discretization of the problem. Error estimates of the semi-discrete and fully-discrete problems are derived. Few numerical examples are provided to verify the theoretical findings.

Nonlinear vibrations of temperature-dependent FGM cylindrical panels subjected to instantaneous surface heating

An analysis is performed in this research to deal with the large amplitude vibrations caused by rapid surface heating in an FGM cylindrical panel. It is assumed that the properties of the panel are distributed through the thickness in terms of volume fraction of constituents. Following the first-order shear deformation theory, von Karman type of kinematics, and Donnell kinematic assumptions, definition of kinetic and strain energies of the shell are established. These obtained equations are discretized via the method of Ritz, where the shape functions are estimated by the Chebyshev polynomials. Temperature distribution within the body is also obtained using the one-dimensional heat conduction equation. This equation is discretized using the finite different method and solved iteratively following the Picard and Crank-Nicolson methods. Thermally induced force and moment resultants are evaluated and inserted into the matrix representation of equations of motion. The temporal evolution of displacement is obtained using the iterative Picard method and Newmark time marching method. Results of this study are compared with the available data in the open literature and after that novel results are given to explore the effects of geometrical, mechanical, and thermal parameters. It is highlighted that power law index, shallowness, thickness, edge supports, temperature dependency, geometrical nonlinearity and thermal boundary conditions are all important factors on the response of the structure under thermal shock. It is highlighted that thermally induced vibrations indeed exists especially in thin class of shells. Besides, thermally induced deflections may be controlled through using a proper power law index.

Phase-field based modeling and simulation for selective laser melting techniques in additive manufacturing

In this study, we develop a phase-field model to describe the solid–liquid phase changes, heat conduction phenomena, during the selective laser melting process. This model is based on the variational principle of minimizing the free energy functional. The proposed model integrates the phase-field equation and the energy equation, which are used to capture the dynamical behavior of the interfacial evolution. We use the semi-implicit Crank–Nicolson scheme and central difference to ensure second-order accuracy in time and space. The numerical scheme is unconditional energy stable. This paper rigorously proves the energy stability of the phase-field model of the Selective Laser Melting process, which confirms the numerical stability and the physical rationality of the solution. Various numerical experiments are performed to verify the robustness of our proposal model. This model can effectively simulate the energy transfer and shape structure changes of the products during the selective laser melting manufacturing process, which provides a reliable guarantee for predicting and optimizing the quality and performance of the selective laser melting process additive manufacturing process.

DiffSim: a user-friendly tool for precise magmatic timescale determination and error propagation via major element diffusion chronometry in magmatic phases

Diffusion chronometry is a technique gaining interest in the scientific community related to volcanology and petrology; however, modelling can be challenging for non-expert users. Here, we present DiffSim, a user-friendly standalone freeware that allows users to calculate magmatic timescales simulating 1D diffusion of major elements in olivine, orthopyroxene, titanomagnetite, and melt (inclusions). The freeware works solving the Fick’s second law equation (for both Cartesian and spherical polar coordinates, depending on the phase) using finite differences through the Crank-Nicolson method. Users must specify the initial composition vs. distance profile, the time resolution, and the intensive conditions (such as temperature, pressure, and oxygen fugacity). For orthorhombic phases, such as olivine and orthopyroxene, users must also specify the plunge and the trend of the (001)-axis and the angle traverse of the 2D section being studied. The best-fitting profile, comparing the natural (measured) and the modelled (calculated) profiles, is obtained using the least-squares fitting method in accordance with the total time specified by the user for performing the diffusion modelling. To determine the uncertainties of the timescale calculation, DiffSim propagates errors based on the uncertainties associated with each intensive condition and the experimental diffusivity measurements. DiffSim is available as executable freeware, allowing researchers and students to use diffusion chronometry to elucidate information about crustal processes with ease and precision.

Numerical integration method for two-parameter singularly perturbed time delay parabolic problem

This study presents an (ε, μ)−uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank–Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

A uniformly convergent method for two-parameter singularly perturbed parabolic partial differential equations with a large shift

A class of singularly perturbed two-parameter parabolic partial differential equations with a large shift is studied. These equations include time-dependent variables and exhibit sensitivity to small perturbations in the system parameters. Moreover, the shift captures the influence of neighbouring states or events on the current evolution of the system. It adds memory-like behaviour to the problem, making their analysis and numerical solution more intricate. The solution of these equations exhibits a multiscale character. The interior and boundary layers exist across which the solution has a steep gradient. A higher-order accurate numerical method is proposed on a non-uniform mesh to solve the problem. The method combines an upwind difference scheme in space and a Crank-Nicolson scheme in time. The method is unconditionally stable, and parameters uniformly convergent with second-order accuracy in time and first-order accuracy in the space variable. Besides, the paper presents numerical results and illustrations to support theoretical estimates.

Collective excitation of quantum droplet with different ranges of the interaction of Pöschl-Teller potential

In this article, we studied quantum droplet with the Pöschl-Teller (PT) interaction potential between the Bose atoms. The Gross–Pitaevskii (GP) equation governs the system. The range and strength of the PT interaction can be adjusted. First, we studied the quantum droplet’s density variation for various PT interaction parameters by the imaginary-time split-step Crank-Nicolson (CN) method. We then used the Bogoliubov theory to examine the collective excitation spectra. We observed that sharp roton forms and phonon modes are missing during long-range interactions. There is a gap at the zero momentum zone due to the long-range PT interaction, which increases with the range and strength of the interaction.

A comparison of Finite difference schemes to Finite volume scheme for axially symmetric 2D heat equation

In this work, we compare finite difference schemes to finite volume scheme for axially symmetric 2D heat equation with Dirichlet and Neumann boundary conditions. Using cylindrical coordinate geometry, we describe a mathematical model of axially symmetric heat conduction for a stationary, homogeneous isotrophic solid with uniform thermal conductivity in a hollow cylinder with an exact solution in a particular case. We obtain the numerical solution of the PDE adapting finite difference and finite volume discretization techniques. Compared to the exact solution, we explore that the numerical schemes are the sufficient tools for the solution of linear or nonlinear PDE with prescribed boundary conditions. Furthermore, the numerical solution discrepancies in the results obtained from Explicit, Implicit and Crank-Nicolson schemes in Finite Difference Method (FDM) are extremely close to the exact solution in the case of Dirichlet boundary condition. The solution from the Explicit scheme is slightly far from the exactsolution and the solutions from Implicit and Crank-Nicolson schemes are extremely close to the exact solution in the case of Neumann boundary condition. Likewise, the numerical solutions obtained in the Finite volume method (FVM) are extremely close to the exact solution in the case of the Dirichlet boundary condition and slightly away from the exact solution in the case of the Neumann boundary condition.

1D analytic and numerical analysis of thin film heat transfer gauges and infra-red cameras in non-planar applications

The impulse response method is widely used for heat transfer analysis in turbomachinery applications. Traditionally, the 1D method assumes a: linear time invariant, isotropic, semi-infinite block with planar surfaces and does not accurately model the true geometric behaviour. This paper evaluates the error introduced by the planar assumption and outlines the required modifications for accurate freeform surface analysis. Adapted cylindrical basis functions are defined for the impulse response method and used to evaluate the impact of the 1D planar assumption. The analytic solutions for both convex and concave surfaces are presented. A penta-diagonal algorithm, for a modified numerical Crank-Nicolson scheme, is also evaluated for fast stable implementation of curved geometry simulation. The scheme shows comparable performance to the impulse response, whilst removing the requirement for linear time invariance. The new methods are demonstrated in the case of aerothermal analysis for heat transfer in a turbine nozzle guide vane. A 3D ANSYS simulation is used as a benchmark and further highlights the importance of the curvature effects. The methods are extended using curvature mapping for infra-red camera data, enabling direct 3D thermal simulation or the fast calculation of freeform curvature. This paper defines the methodology to analyse heat transfer measurements on non-planar geometry.

Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions

Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Herein, we mainly develop a new reduced-dimension method for the unknown finite element (FE) solution coefficient vectors to the elastic wave equation containing the known singular source term, the unknown displacement vector, and the unknown stress tensor matrix. For this end, we need first to develop the Crank-Nicolson (CN) FE (CNFE) method with unconditional stability, unconditional convergence, and second-order time accuracy for the elastic wave equation and analyze the existence, unconditional stability, and errors of the CNFE solutions. After that we employ a proper orthogonal decomposition to lower the dimensionality of the unknown FE solution coefficient vectors in the CNFE method for the elastic wave equation and to design a new reduced-dimension extrapolated CNFE (RDECNFE) method. Subsequently, we employ matrix analysis to discuss the existence, unconditional stability, unconditional convergence, and errors of the RDECNFE solutions. Finally, we use two numerical examples to show the superiority of the RDECNFE method and our theoretical results. It is also shown that the RDECNFE method is feasible and effective for solving the elastic wave equation.

Physical invariants-preserving compact difference schemes for the coupled nonlinear Schrödinger-KdV equations

In this paper, we develop efficient compact difference schemes for the coupled nonlinear Schrödinger-KdV (CNLS-KdV) equations to conserve all the physical invariants, namely, the energy of oscillations, the number of plasmon, the number of particle and the momentum. By combining the exponential scalar auxiliary variable (E-SAV) approach, we reconstruct the original CNLS-KdV equations and adopt the compact difference method and Crank-Nicolson method to develop energy stable scheme. The E-SAV compact difference scheme preserves the total energy and the number of particle. We further introduce two Lagrange multipliers for the E-SAV reformulation system to develop compact difference scheme, which preserves exactly the number of plasmon and the momentum. At each time step for the second scheme, we only need to solve linear systems with constant coefficients and nonlinear quadratic algebraic equations which can be efficiently solved by Newton's iteration. Numerical experiments are given to show the effectiveness, accuracy and performance of the proposed schemes.

A Modified Crank-Nicolson Method for Numerical Solution of the Linear Time-Dependent Schrodinger Equation

This paper presents the effectiveness of the solution of the time-dependent Schrodinger equation using the modified Crank-Nicolson Method (MCNM). The method was developed through finite difference approximation. To demonstrate the effectiveness of this approach, three different problems were tested with these approaches by using MATLAB soft code. Also, this research verifies that, in the presence or absence of potential energy, the modified Crank-Nicolson method for solving the time-dependent Schrödinger equation is accurate, convergent, and computationally efficient.

A fast solvable operator-splitting scheme for time-dependent advection diffusion equation

It is well known that the implicit central difference discretization for unsteady advection diffusion equation (ADE) suffers from being time-consuming to solve when the advection term dominates. In this paper, we propose an operator-splitting scheme for the unsteady ADE, in which the ADE is firstly discretized by Crank-Nicolson (CN) scheme in time and central difference scheme in space; and then the discrete advection-diffusion problem is split as advection sub-problem and diffusion sub-problem at each time-level. The significance of the new scheme is that these sub-problems can be fast and directly solved within a linearithmic complexity (a linear-times-logarithm complexity) by means of fast sine transforms (FSTs). In particular, the complexity is independent of the dominance of the advection term. Theoretically, we show that proposed scheme is unconditionally stable and of second-order convergence in time and space. Numerical results are reported to show the efficiency of the proposed scheme.

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

Corium interface flow dynamics investigation during severe accident in pressurized water reactors using compressive advection interface capturing method

Past nuclear accident occurrences raised strong concerns which led to research on nuclear safety. One of the major causes of nuclear accidents is the impeded circulation of core coolant, leading to decay heat removal cessation and rapid temperature rise. If uncontrolled, this results in critical heat flux, loss of coolant accidents, and core dryout. Detailed melted core relocation (i.e., nuclear fuel, graphite, and zircaloy) needs to be investigated through interface capture and multimaterial flow model coupling, which have not been done in previous studies. This work aims to investigate the impacts of temperature and core material composition on the flow dynamics during core relocation. In this study, mass fraction is discretized using a streamlined upwind Petrov–Galerkin method spatially and a modified Crank–Nicolson method temporally to accurately capture fluid interfaces using a high-order accurate flux-limiter. Two core material composition cases (individual material properties case and bulk material properties case) were considered to assess the impact of temperature and core materials composition on both flow dynamics and computational time. Temperature has a significant impact on core material transport and corium flow dynamics during core relocation. Bulk materials properties case has greater impact of temperature on its corium resulting in faster materials transport, but with higher computation time.

A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility

A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility