Nicolson Method Research Articles

Semi-implicit Solver for the Heat Equation with Stefan–Boltzmann Law Boundary Condition

The surface energy balance on an atmosphereless body consists of solar irradiance, subsurface heat conduction, and thermal radiation to space by the Stefan–Boltzmann law. Here we extend the semi-implicit Crank–Nicolson method to this specific nonlinear boundary condition and validate its accuracy. A rapid change in incoming solar flux can cause a numerical instability, and several approaches to dampen this instability are analyzed. A predictor based on the Volterra integral equation formulation for the heat equation is also derived and can be used to improve accuracy and stability. The publicly available implementation provides a fast and robust thermophysical model that has been applied to lunar, Martian, and asteroidal surfaces, on occasion to millions of surface facets or parameter combinations.

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

This article is concerned with a class of singularly perturbed semilinear parabolic convection‐diffusion partial differential equations (PDEs) with discontinuous source function. Solutions of these PDEs usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. We begin our study by proving existence of the analytical solution of the considered nonlinear PDE by means of the upper and lower solutions approach; and the ‐uniform stability of the analytical solution is established by using the comparison principle for the continuous nonlinear operator. In order to realize the asymptotic behavior of the analytical solution, we derive a priori bounds of the solution derivatives via decomposition of the solution into the smooth and the layer components. For an efficient numerical solution of the nonlinear PDE, the time‐derivative is approximated by the Crank–Nicolson method on an equidistant mesh, and we approximate the spatial derivative by a finite difference scheme on a suitable layer‐adapted mesh. We establish the comparison principle for the nonlinear difference operator to prove the ‐uniform stability of the discrete solution and further construct a suitable decomposition of the discrete solution for pursuing the convergence analysis. The computational method is proven to be parameter‐robust with second‐order time accuracy in the discrete supremum norm. The theoretical estimate is finally verified by the numerical experiments.

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

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.

Analysis and application of MLPG7 for diffusion equations with nonlinear reaction terms

This paper extends the recently proposed variant of meshless local Petrov Galerkin (MLPG) method, i.e., MLPG7, for solving time dependent PDEs. As test function, the method uses a novel modification of fundamental solution of Laplace operator that not only the test function itself but also its derivative vanish on boundary of local subdomains. Therefore, more stable local integral equations are obtained by replacing a boundary integral of derivatives of field variables with a domain integral of field variables itself. MLPG7 formulation converts the nonlinear governing equation into a system of first order ordinary differential equations (ODEs). For this system of ODEs, a theoretical stability analysis is carried out for the case of regularly spaced nodal points. Stability and convergence conditions for fully discretized equations by Euler (forward and backward) and Crank–Nicolson methods are investigated. The nonlinear term is treated iteratively. A numerical study investigates the convergence and stability of the method. A comparison is also done with some well-known methods and the results reveal the excellence of MLPG7. The convergence of the iterative approach is numerically verified by another illustrative test example. As nonlinear test problems, Allen–Cahn equations are studied.

Iterated Crank–Nicolson Method for Peridynamic Models

In this paper, we explore the iterated Crank–Nicolson (ICN) algorithm for the one-dimensional peridynamic model. The peridynamic equation of motion is an integro-differential equation that governs structural deformations such as fractures. The ICN method was originally developed for hyperbolic advection equations. In peridynamics, we apply the ICN algorithm for temporal discretization and the midpoint quadrature method for spatial integration. Several numerical tests are carried out to evaluate the performance of the ICN method. In general, the ICN method demonstrates second-order accuracy, consistent with the Störmer–Verlet (SV) method. When the weight is 1/3, the ICN method behaves as a third-order Runge–Kutta method and maintains strong stability-preserving (SSP) properties for linear problems. Regarding energy conservation, the ICN algorithm maintains at least second-order accuracy, making it superior to the SV method, which converges linearly. Furthermore, selecting a weight of 0.25 results in fourth-order superconvergent energy variation for the ICN method. In this case, the ICN method exhibits energy variation similar to that of the fourth-order Runge–Kutta method but operates approximately 20% faster. Higher-order convergence for energy can also be achieved by increasing the number of iterations in the ICN method.

Stabilization of hollow Gaussian beams in nonlinear metamaterial waveguides

Hollow Gaussian beams which possess the dark hollow spatial intensity distribution were found to have interesting applications in trapping nano-sized particles and free-space optical communication. Many applications demand stable dynamics of higher dimensional beams. Here we theoretically study the dynamics of hollow Gaussian beams in cubic and quintic nonlinear negative index metamaterial. We adopt Lagrangian variational analysis and explore the parametric regions characterizing the metamaterial in which the hollow Gaussian beam can execute stable dynamics as well as it can undergo filamentation due to intrinsic collapse. We confirm the analytical results obtained through variational analysis by direct numerical analysis based Crank Nicolson method implemented in Python programming.

Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells Under Instantaneous Heating

This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the [Formula: see text]-Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

A fully implicit edge/face centered discontinuous Galerkin / mixed finite element scheme for the advection-dispersion equation

Mixed Finite Element (MFE) method is a robust numerical technique for solving elliptic and parabolic partial differential equations (PDEs). However, MFE can generate solutions with strong unphysical oscillations and/or large numerical diffusion for hyperbolic type PDEs. For its part, Discontinuous Galerkin (DG) finite element method is well adapted to solve hyperbolic systems and can accurately reproduce solutions involving sharp fronts. Therefore, the combination of DG and MFE is a good strategy for solving hyperbolic/parabolic problems such as advection – diffusion/dispersion equations. The classical formulation of the two methods is based on operator and time splitting allowing for separate solutions to advection with an explicit scheme and to dispersion with an implicit scheme. However, this kind of approach has the following drawbacks: (i) it lacks efficiency, as two systems with different unknowns are solved at each time step, (ii) it induces errors generated by the splitting, (iii) it can be CPU wise-expensive because of the CFL constraint, and (iv) it cannot be employed for steady-state transport simulations.To overcome these difficulties, we develop in this work a fully implicit edge/face centered DG-MFE formulation where the two methods share the same unknowns. In this formulation, the DG method is developed on lumping regions associated with the mesh edges/faces instead of mesh elements. Thus, the traces of concentration at mesh edges/faces, which are the Degrees Of Freedom (DOF) of the hybrid-MFE, are also part of the DOFs of the DG. The temporal discretization is based on the Crank–Nicolson method for both advection and dispersion. Numerical tests are performed to validate the new scheme by comparison against an analytical solution and to show its ability to handle steady-state transport simulations.The procedure is developed for 2D triangular meshes but can easily be extended to other 2D and 3D shape elements.

Investigation the Accuracy of Crank Nicolson Methods and their Modified Schemes for One-Dimensional Linear Convection-Reaction-Diffusion Equations

In this paper, Crank-Nicolson and Their Modified scheme are applied to find the solution of the convection-reaction-diffusion equation. First, the given solution domain is discretized. Then, using Taylor series expansion, we obtain the difference scheme of the model problem. By rearranging this scheme, we gain proposed techniques. To verify the validity of the proposed techniques, two model illustrations are considered. The stability and convergent analysis of the present scheme is worked by supporting the theoretical and numerical error bound. The accuracy of the present scheme has been shown in the sense of average absolute error, root means square error, and maximum absolute error norms. Then, the accuracy of the present techniques is compared with the accuracy obtained by another method in the literature. The physical behavior of the present results also has been shown in terms of graphs. As we can seen from the results in tables and physical behavior of solution, the present scheme approximates the exact solution veritably well. These scheme is improve the approximation exact solution of convection-reaction-diffusion equation. Hence, the present solution of convection-reaction-diffusion equation is accurate than pre-existing solution.

Crank–Nicolson method for solving time-fractional singularly perturbed delay partial differential equations

Crank–Nicolson method for solving time-fractional singularly perturbed delay partial differential equations

Triply periodic minimal surfaces based topology optimization for the hydrodynamic and convective heat transfer

Triply periodic minimal surfaces (TPMS), possessing diverse and crucial properties for thermal convection analysis, hold significant promise for topology optimization in convective and hydrodynamic systems. This study aims to develop an innovative topology optimization method based on TPMS for channel shape design. We focus on maximizing heat dissipation and mass flux while maintaining a constant mean curvature. The channel structures undergo free evolution during the optimization process, resulting in efficient and intricate geometries based on the TPMS assumption. To incorporate TPMS properties, we modify the original energy formulation, which comprises kinetic energy, thermal energy, and Ginzburg–Landau energy. The governing system encompasses the phase field model, the Darcy–Stokes model, and the reaction–diffusion heat transfer model. The hydrodynamic response within the phase change structure is computed by solving the transient Darcy–Stokes equation augmented with a temperature-dependent diffusion term. To achieve second-order temporal and spatial accuracy, we employ the Crank–Nicolson method for the time scale and the central difference method for the spatial scale. Through the utilization of the Lagrange multiplier method, we establish the unconditional decrease of the discretized original energy. This implies that the proposed scheme can utilize a large time step. To verify the robustness of our proposed method, we conduct several numerical tests from both qualitative and quantitative perspectives. The results demonstrate the effectiveness of our method in enhancing heat and mass transfer capabilities.

Conservative higher-order finite difference scheme for the coupled nonlinear Schrödinger equations

This paper introduces a conservative higher-order finite difference scheme for solving the coupled nonlinear Schrödinger equations. The Crank–Nicolson method is employed to discretize time derivatives and the sixth-order difference operator is used to discretize space derivatives, correspondingly, the resulting difference scheme has second-order accuracy in time and sixth-order accuracy in space. By utilizing the discrete energy method, the conservation of discrete mass and energy, the boundedness, existence and uniqueness of solution, unconditional stability and the convergence of the new scheme are proved. Then making use of Richardson extrapolation, the time accuracy is increased to the fourth order. Finally, the numerical experiments are conducted to validate the theoretical results presented in the paper.

Finite Difference Modeling of Time Fractal Impact on Unsteady Magneto-hydrodynamic Darcy–Forchheimer Flow in Non-Newtonian Nanofluids with the q-Derivative

This contribution addresses a fractal numerical scheme that can be employed for handling fractal time-dependent parabolic equations. The numerical scheme presented in this contribution can be used to discretize integer order and fractal derivatives in a given differential equation. Therefore, the scheme and results can be used for both cases. The proposed finite difference scheme is based on two stages. Fractal time derivatives are discretized by employing the proposed approach. For the scalar convection–diffusion equation, we derive the stability condition of the proposed fractal scheme. Using a nonlinear chemical reaction, the approach is also used to solve the Quantum Calculus model of a Williamson nanofluid’s unsteady Darcy–Forchheimer flow over flat and oscillatory sheets. The findings indicate a negative correlation between the velocity profile and the porosity parameter and inertia coefficient, with an increase in these factors resulting in a drop in the velocity profile. Additionally, the fractal scheme under consideration is being compared to the fractal Crank–Nicolson method, revealing that the proposed scheme exhibits a superior convergence speed compared to the fractal Crank–Nicolson method. Several problems involving the motion of non-Newtonian nanofluids through magnetic fields and porous media can be investigated with the help of the proposed numerical scheme. This research has implications for developing more efficient heat transfer and energy conversion devices based on nanofluids.

A comparison of Rosenbrock–Wanner and Crank–Nicolson time integrators for atmospheric modelling

AbstractNonhydrostatic atmospheric models often use semi‐implicit temporal discretisations in order to negate the time‐step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision using Newton's method is considered prohibitively expensive, and so the nonlinear solver is typically truncated to a fixed number of iterations, often using an approximate Jacobian matrix that is reassembled only once per time step. The present article studies the impact of using various third‐order, four stage Rosenbrock–Wanner schemes, where integration weights are chosen to meet specific stability and order conditions, in comparison to a Crank–Nicolson time discretisation, as is done in the UK Met Office's LFRic model. Rosenbrock–Wanner schemes present a promising alternative on account of their ability to preserve their temporal order with only an approximate Jacobian, and may be constructed to be stiffly‐stable, so as to ensure the decay of fast unresolved modes. These schemes are compared for the 2D rotating shallow‐water equations and the 3D compressible Euler equations at both planetary and nonhydrostatic scales and are shown to exhibit improved results in terms of their energetic profiles and stability. Results in terms of computational performance are mixed, with the Crank–Nicolson method allowing for longer time steps and faster time to solution for the baroclinic instability test case at planetary scales, and the Rosenbrock–Wanner methods allowing for longer time steps and faster time to solution for a rising bubble test case at nonhydrostatic scales.

Significance of Cu-Fe3O4 on fractional Maxwell fluid flow over a cone with Newtonian heating

The goal of this research is to investigate fractional Maxwell hybrid nanofluids utilizing partial differential equations in terms of Caputo time fractional derivatives. Specifically, the effect of Newtonian heating on the thermal performance of a fractional Maxwell hybrid nanofluid moving over a permeable cone in the presence of thermal radiation and heat generation is considered. The Crank–Nicolson method and L1 algorithmt of Caputo derivative are used to find numerical solutions to the considered nonlinear problem. The effects of significant flow factors on fluid properties are examined and illustrated in various graphs. According to the results, the thermal performance of the fluid raised by 0.4%, 6.1%, and 3.1% on adding 4% volume fraction of Fe3O4, Cu, and Cu−Fe3O4, respectively, in the base fluid.

Reduction of phosphorus diffusion in bulk germanium via argon/phosphorus co-implantation and RTA annealing

Germanium has received increased research interest for use in next-generation CMOS technology as its high carrier mobilities allow for enhanced device performance without further device scaling. Fabrication of high-performance NMOS Ge devices is hindered by high diffusivity and low activation of n-type implanted dopants. While the high solid solubility of P in Ge makes it an ideal dopant, its diffusion mechanism is poorly understood and results in heavy tradeoffs between implanted dopant diffusion and electrical activation. In this study, we demonstrate the suppression of in-diffusion of implanted P via a co-implantation with Ar. Diffusivity of implanted P species and their activation is investigated over a wide range of annealing temperatures and times. P diffusion was explored by secondary-ion-mass-spectrometry and the diffusivity of P was extracted by solving the 2D diffusion equation using the Crank–Nicolson method, and the dopant electrical activation was extracted from the Hall effect measurements. The co-implantation of P with Ar entirely suppresses P in-diffusion up to annealing temperatures as high as 700 °C but at the cost of its reduced electrical activation. Extracted diffusivity reveals a highly correlated exponential relationship with annealing. P activation energy was extracted from Arrhenius behavior. A 450 °C/10 min annealing of P implant shows negligible in-diffusion of P with the activation as high as 70%. RTA processing of the Ar/P co-implanted sample at 750 °C for 1 min results in a negligible P in-diffusion and an electrical activation of 20%.

Optimal time separation modeling and simulation for stable neon isotopes in a transient square cascade: a comprehensive study

This study simulates the separation of stable Neon isotopes in a transient square cascade using the Crank–Nicolson method and the q-iteration method. The simulations show that each isotope has an optimal value in either the product or waste streams at a specific time, which affects the cost of separation. The results demonstrate that the optimal separation time for Ne-20 increases with the feed concentration and is influenced by the unit separation factor. Depending on the feed concentration and separation parameter, Ne-21 may be enriched in either the product or waste stream. The optimal separation time for Ne-22 is the maximum operation time and is not affected by changes in feed concentration or unit separation factor. Calculating the optimal separation time is crucial for obtaining target isotopes with specific purity at a designated time, depending on their intended applications.

Numerical analysis of temporal second‐order accurate scheme for the abstract Volterra integrodifferential equation

In this paper, a second‐order accurate scheme is considered for the temporal discretization of the abstract Volterra integrodifferential equation with a weakly singular kernel. The time discrete scheme is constructed by the Crank–Nicolson method for approximating the time derivative and product integration (PI) rule for approximating the integral term. The proposed scheme employs a graded mesh for time to compensate for the singular behavior of the exact solution at . Under the suitable assumptions, the stability and convergence are established by the energy argument, and the error is of order , where is the parameter for the time grids. Numerical experiments validate the theoretical estimate.

Stability, convergence and error analysis of B-spline collocation with Crank–Nicolson method and finite element methods for numerical solution of Schrodinger equation arises in quantum mechanics

In the present study, the complex-valued Schrodinger equation (CVSE) is solved numerically by a nonic B-spline finite element method (FEM) in quantum mechanics. The approach employed is based on the collocation of nonic B-splines over spatial finite elements, so that we have continuity of the dependent variable and its first eight derivatives throughout the solution range. For time discretization, the Crank-Nicolson scheme of second order based on FEM is employed. The method is shown to be unconditionally stable and accurate to order. Three problems are considered to validate the algorithm. Comparisons are made with existing methods and analytical solutions. The proposed method exhibits good conservation properties and performs well with regards to analytical solutions for different error norms and conservative constants related to parameters in quantum classes in Physics. The computational complexity of (2N+18) arithmetic operations with the help of a nonic-diagonal matrix is also tackled by the present scheme.