Space Step Size Research Articles

Application of modified iterative method to simulate rainfall infiltration in unsaturated soils

A modified Gauss-Seidel iterative method (MP(m)-GS) with multistep preconditioner is developed to solve partial differential equations of rainfall infiltration. The finite difference method (FDM) is applied to numerical discretization. Furthermore, a system of linear equations is solved using an iterative scheme. For some unfavorable numerical conditions, such as large initial conditions, classical Picard iterative method usually has low accuracy and computational efficiency. Conventional linear iterative methods have a slower convergence rate, such as Gauss-Seidel (GS) and Jacobi iterative methods, particularly for small discrete space step size. Thus, MP(m)-GS is developed to simulate rainfall infiltration into unsaturated soils. The analytical solutions are employed to verify the proposed methods. The results indicate that the MP(m)-GS has higher computational efficiency and accuracy than the improved Picard methods. Compared with the conventional linear iterative methods GS and SOR, MP(m)-GS also demonstrates faster convergence rate and higher computation efficiency. The results show nice applications in modeling rainfall infiltration in unsaturated soils.

Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations

In this paper, we construct a high order difference scheme for two-dimensional semilinear fractional sub-diffusion equations at first. To reduce the computation time, an efficient time two-grid algorithm is then proposed. The global convergence order of the two-grid scheme reaches O(τF2+τC4+h12+h22), where τF and τC represent the time-step sizes on the fine and coarse grids respectively, while h1 and h2 are the space-step sizes. Furthermore, stability and convergence of the scheme are carefully studied by some skills. At last, the theoretical statements are verified by numerical experiments.

The Convergence of Attractors for Some Discrete Cahn‐Hilliard Systems

In this article, we use a finite difference scheme to discretize the Cahn‐Hilliard equation with the space step size h. We first prove that this semidiscrete system inherits two important properties, called the conservation of mass and the decrease of the total energy, from the original equation. Then, we show that the semidiscrete system has an attractor on a subspace of ℝN+1. Finally, the convergence of attractors is established as the space step size h of the semidiscrete Cahn‐Hilliard equation tends to 0.

A high order numerical technique and its analysis for nonlinear generalized Fisher’s equation

This paper deals with the design and analysis of a high order numerical scheme for the nonlinear time-fractional generalized Fisher’s equation (TFGFE). The Caputo fractional derivative (FD) of order α, (α∈(0,1)) appearing in the model problem is approximated by means of L1−2 scheme. The discretization for the space derivative is made by a collocation method based on quintic B-spline (QBS) basis function. Convergence analysis of the method is established. Five examples are provided to demonstrate the efficiency and feasibility of the method. The influence of α on the solution profile of the TFGFE is examined. It is shown that our method is of O(Δt2+Δx4) accuracy, where Δt and Δx respectively represent the time and space step sizes. The results obtained are compared with those of other three methods. The CPU time (in seconds) is given in order to justify the computational efficiency of proposed numerical scheme.

An improved Picard iteration scheme for simulating unsaturated flow in porous media

The Richards equation is widely used in the simulation of unsaturated flow and related fields. In the numerical solution process, the finite volume method can be used to carry out numerical discretization, and the Picard method is used for iterative solution. However, in order to obtain a more reliable numerical solution, the space step size of the conventional uniform grid is often small. Especially under some unfavorable numerical conditions, such as infiltration into dry soil and layered soil with very different hydraulic conductivity, this will be very time-consuming and even slow to converge. Thus, combined with the non-uniform grid in the form of Chebyshev, an improved Picard method based on the non-uniform two-grid correction scheme (NTG-PI) is proposed to model 1D unsaturated flow in porous media. Through three examples of unsaturated flow under unfavorable conditions, the proposed scheme was verified. The results show that, compared with the conventional Picard method, NTG-PI can obtain higher numerical accuracy with a smaller number of nodes, and also has a faster convergence rate. This method can provide a certain reference for the numerical simulation of unsaturated flow.

A high-order scheme for time-space fractional diffusion equations with Caputo-Riesz derivatives

In this paper, we present a high-order approach for solving one- and two-dimensional time-space fractional diffusion equations (FDEs) with Caputo-Riesz derivatives. To design the scheme, the Caputo temporal derivative is approximated using a high-order method, and the spatial Riesz derivative is discretized by the second-order weighted and shifted Grünwald difference (WSGD) method. It is proved that the scheme is unconditionally stable and convergent with the order of O(ταh2+τ4), where τ and h are time and space step sizes, respectively. We illustrate the accuracy and effectiveness of the method by providing several numerical examples.

Interpreting SAXS data recorded on cellulose rich pulps

A simulation method was developed for modelling SAXS data recorded on cellulose rich pulps. The modelling method is independent of the establishment of separate form factors and structure factors and was used to model SAXS data recorded on dense samples. An advantage of the modelling method is that it made it possible to connect experimental SAXS data to apparent average sizes of particles and cavities at different sample solid contents. Experimental SAXS data could be modelled as a superposition of a limited number of simulated intensity components and gave results in qualitative agreement with CP/MAS 13C-NMR data recorded on the same samples. For the water swollen samples, results obtained by the SAXS modelling method and results obtained from CP/MAS 13C-NMR measurements, agreed on the ranking of particle sizes in the different samples. The SAXS modelling method is dependent on simulations of autocorrelation functions and the time needed for simulations could be reduced by rescaling of simulated correlation functions due to their independence of the choice of step size in real space. In this way an autocorrelation function simulated for a specific sample could be used to generate SAXS intensity profiles corresponding to all length scales for that sample and used for efficient modelling of the experimental data recorded on that sample.Graphical abstract

A second order linear energy stable numerical method for the Cahn–Hilliard–Hele–Shaw system

In this paper, we consider the numerical approximations for the Cahn–Hilliard–Hele–Shaw system, which is a modified Cahn–Hilliard equation coupled with the Darcy flow law. One of challenges in solving this system numerically is how to develop linear discretization for the nonlinear term, while preserving the energy stability. By introducing a Lagrange multiplier r, we construct a fully discrete, second-order linear scheme. The proposed scheme is rigorously proven to be uniquely solvable and unconditionally stable in energy. Moreover, we show theoretical analysis on error estimates of the time step size τ and space step size h at the discrete level. Several numerical examples are presented to demonstrate the stability, accuracy and efficiency of the proposed scheme. We also show numerical simulations on the coarsening dynamics.

An unconditionally stable second-order linear scheme for the Cahn-Hilliard-Hele-Shaw system

In this paper, we focus on the numerical approximation of the Cahn-Hilliard-Hele-Shaw system. Firstly, based on the idea of the stabilized method, an unconditionally stable linear scheme with second-order accuracy in time and space is proposed, which is modified from the Crank-Nicolson scheme. Secondly, we derive that the proposed numerical scheme is unconditionally stable, without any restriction for the time step size. After a careful calculation, we get discrete error estimates of the time step size τ and space step size h. Finally, numerical simulations of energy dissipation and spinodal decomposition are presented to demonstrate the stability, accuracy and efficiency of the proposed scheme.

A linearized compact ADI numerical method for the two-dimensional nonlinear delayed Schrödinger equation

We develop a linearized compact alternating direction implicit (ADI) numerical method to solve the nonlinear delayed Schrödinger equation in two-dimensional space. By discrete energy estimate method, we analyse the convergence of the fully-discrete numerical method, and show that the numerical scheme is of order O(Δt2+h4) with time stepsize Δt and space stepsize h. At last, we present several numerical examples to confirm theoretical analyses.

Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation

The numerical approximation for the Landau-Lifshitz equation, which models the dynamics of the magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting challenges in developing numerical methods. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz equation based on the second-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. Subsequently, we provide a rigorous convergence analysis for the fully discrete numerical solution by the introduction of two sets of approximated solutions where one set of solutions solves the Landau-Lifshitz equation and the other is projected onto the unit sphere. Second-order accuracy in both time and space is obtained provided that the spatial step-size is the same order as the temporal step-size. And also, the unique solvability of the numerical solution without any assumption for the step-size in both time and space is theoretically justified, using a monotonicity analysis. All these theoretical properties are verified by numerical examples in both 1D and 3D spaces.

High-Order Numerical Method for Solving a Space Distributed-Order Time-Fractional Diffusion Equation

This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation. First, we use the mid-point quadrature rule to transform the space distributed-order term into multi-term fractional derivatives. Second, based on the piecewise-quadratic polynomials, we construct the nodal basis functions, and then discretize the multi-term fractional equation by the finite volume method. For the time-fractional derivative, the finite difference method is used. Finally, the iterative scheme is proved to be unconditionally stable and convergent with the accuracy O(σ2 + τ2−β + h3), where τ and h are the time step size and the space step size, respectively. A numerical example is presented to verify the effectiveness of the proposed method.

Option Pricing by the Legendre Wavelets Method

This paper presents the numerical solution of the Black–Scholes partial differential equation (PDE) for the evaluation of European call and put options. The proposed method is based on the finite difference and Legendre wavelets aproximation scheme. We derive a matrix structure for the Legendre wavelets integral operator which has been widely used so far. Moreover, in order to use the payoff function, another operational matrix is derived. By the proposed combined method, the solving Black–Scholes PDE problem reduces to those of solving a Sylvester equation. The proposed algorithms show that in compared to literature methods, the proposed method is easy to be implemented and have high execution speed. Furthermore, we prove that the obtained Sylvester equation has a unique solution. In addition, the effect of the finite difference space step size to the computational accuracy is studied. For having suitable solution, the numerical solutions show that there is no need to select very small step size. Also only a small number of basis functions in the Legendre wavelets series is needed. The numerical results demonstrate efficiency and capability of the proposed method.

L 1 -Multiscale Galerkin’s Scheme with Multilevel Augmentation Algorithm for Solving Time Fractional Burgers’ Equation

In this paper, we consider the initial boundary value problem of the time fractional Burgers equation. A fully discrete scheme is proposed for the time fractional nonlinear Burgers equation with time discretized by L 1 -type formula and space discretized by the multiscale Galerkin method. The optimal convergence orders reach O τ 2 − α + h r in the L 2 norm and O τ 2 − α + h r − 1 in the H 1 norm, respectively, in which τ is the time step size, h is the space step size, and r is the order of piecewise polynomial space. Then, a fast multilevel augmentation method (MAM) is developed for solving the nonlinear algebraic equations resulting from the fully discrete scheme at each time step. We show that the MAM preserves the optimal convergence orders, and the computational cost is greatly reduced. Numerical experiments are presented to verify the theoretical analysis, and comparisons between MAM and Newton’s method show the efficiency of our algorithm.

Dynamic model of cable tension and configuration for vessel at anchor

A numerical model considering the seabed friction is developed to investigate the dynamic behavior of anchor cable in this paper. In the model, the anchor cable was divided into hanging and lying parts. The governing equations of hanging and lying anchor cables were established respectively based on finite difference method and coupled according to the boundary conditions. The coupled equations of anchor cable were solved by Newton iteration and the static anchor cable parameters used for the initial values were confirmed by double bisection method. To illustrate the reasonability and validity of the model, several simulations of vessel sway in wind with different time and space step sizes and different seabed friction coefficients were presented. The simulation results show that anchor cable snap load is 3–5 times of average tension, the variation of anchor cable bearing is 2 times of ship heading, and the sway amplitude of the ship’s center of gravity is about one times of ship length. Compared with the experimental data, the model and algorithm in this paper can successfully provide the realistic prediction of the dynamic tension and configuration of anchor cable, and can be used as a reference for anchoring operation. Farther, this paper found out that the normal seabed friction coefficient has considerable impact on the cable tension.

Time-fractional diffusion equation-based image denoising model

In the last two decades, due to the use of total variation (TV) denoising, variation models for image reconstruction have achieved great success. Compare to some known variation models which have just adopted the space-fractional diffusion equation, the time-fractional diffusion equation has another adjustable time-fractional derivative order to control the diffusion process. Therefore, in this paper, a new denoising model based on time-fractional diffusion equation is proposed, where the discretization in space is the application of classical difference scheme while the discretization in time is the approximation of the Caputo differential. Furthermore, both the explicit and implicit numerical scheme named TFEIS are built and its stability and convergence are rigorously discussed. We prove the full-discretization is stable under some conditions and the numerical solution converges to the exact one with order $$O(\tau^{2 - \alpha } + h^{2} )$$ , where τ and $$h$$ are the time and space step size, respectively. Finally, various evaluation indexes such as histogram recovery degree, signal-to-noise ratio, and edge texture retention index are applied to compare new and original models comprehensively in lots of images. Computer simulation results show our models

A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation

The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an efficient finite difference scheme to approximate the tempered fractional derivatives and then a fully discrete LDG method for the tempered fractional diffusion equation. We prove that the scheme is unconditionally stable and convergent with the order \begin{document}$ O(h^{k+1}+\tau^{2-\alpha}) $\end{document} , where \begin{document}$ h, \tau $\end{document} and \begin{document}$ k $\end{document} are the step size in space, time and the degree of piecewise polynomials, respectively. Finally numerical experimets are performed to show the effectiveness and testify the accuracy of the method.

A mathematical analysis of an isothermal tube drawing process

The aim of this paper is to investigate stability of an isothermal glass tube drawing process through control parameters by the method of linear stability analysis. We want to see how the process parameters effect stability of the physical system being considered. For this purpose, we not only prove the existence and uniqueness of the solutions of steady state isothermal tube drawing model but also determine its numerical solution. To perform linear stability analysis, steady state numerical solution is incorporated in the eigenvalue problem, formulated by linearizing the isothermal model. The eigenvalue problem is then solved numerically to determine the critical draw ratio which indicates the onset of instabilities. To the end, stability of the process is analyzed using three different values of space step size. We also observe and discuss the effect of density, viscosity and pressure on stability of the isothermal tube drawing model.

Local Discontinuous Galerkin Method for Nonlinear Time-Space Fractional Subdiffusion/Superdiffusion Equations

Fractional partial differential equations with time-space fractional derivatives describe some important physical phenomena. For example, the subdiffusion equation (time order 0<α<1) is more suitable to describe the phenomena of charge carrier transport in amorphous semiconductors, nuclear magnetic resonance (NMR) diffusometry in percolative, Rouse, or reptation dynamics in polymeric systems, the diffusion of a scalar tracer in an array of convection rolls, or the dynamics of a bead in a polymeric network, and so on. However, the superdiffusion case (1<α<2) is more accurate to depict the special domains of rotating flows, collective slip diffusion on solid surfaces, layered velocity fields, Richardson turbulent diffusion, bulk-surface exchange controlled dynamics in porous glasses, the transport in micelle systems and heterogeneous rocks, quantum optics, single molecule spectroscopy, the transport in turbulent plasma, bacterial motion, and even for the flight of an albatross (for more physical applications of fractional sub-super diffusion equations, one can see Metzler and Klafter in 2000). In this work, we establish two fully discrete numerical schemes for solving a class of nonlinear time-space fractional subdiffusion/superdiffusion equations by using backward Euler difference 1<α<2 or second-order central difference 1<α<2/local discontinuous Galerkin finite element mixed method. By introducing the mathematical induction method, we show the concrete analysis for the stability and the convergence rate under the L2 norm of the two LDG schemes. In the end, we adopt several numerical experiments to validate the proposed model and demonstrate the features of the two numerical schemes, such as the optimal convergence rate in space direction is close to Ohk+1. The convergence rate in time direction can arrive at Oτ2−α when the fractional derivative is 0<α<1. If the fractional derivative parameter is 1<α<2 and we choose the relationship as h=C′τ (h denotes the space step size, C′ is a constant, and τ is the time step size), then the time convergence rate can reach to Oτ3−α. The experiment results illustrate that the proposed method is effective in solving nonlinear time-space fractional subdiffusion/superdiffusion equations.

Simulation of 3D grinding temperature field by using an improved finite difference method

3D grinding temperature field can be simulated by the finite difference method. When the finite difference method is used to calculate the temperature field, the boundary conditions need to be considered. Unfortunately, the difference equations for internal nodes and boundary nodes are not the same. For cuboid computing domains, there are twenty-six types of boundary nodes. This reduces the calculation efficiency to a certain extent. An improved finite difference method for the 3D grinding temperature field was proposed to solve this problem in this paper. By adding auxiliary nodes, the boundary nodes are converted into internal nodes, and the difference equations of the boundary nodes and internal nodes are unified. In this paper, the improved algorithm was used to simulate the 3D grinding temperature field. The temperature distribution characteristics of the grinding temperature field were analyzed. Then, the effects of heat source type, space step size, and convective heat transfer coefficient on the temperature field were studied. The results show that the improved algorithm can well simulate the 3D grinding temperature field. Finally, compared with the original algorithm, the calculation results were completely consistent, and the efficiency is improved by about 20%.