Second-order Difference Scheme Research Articles

Uniformly convergent second-order difference scheme for a singularly perturbed periodical boundary value problem

A periodic boundary value problem with a small parameter multiplying the first- and second-order derivatives is considered. The problem is discretized using a hybrid difference scheme on a Shishkin mesh. We show that the scheme is almost second-order convergent in the maximum norm, which is independent of a singular perturbation parameter. Numerical experiment supports these theoretical results.

A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation

In this paper, we present a Crank–Nicolson type finite difference scheme to approximate the nonlinear evolutionary Extended Fisher–Kolmogorov (EFK) equation. We prove the existence of the solution by using the well-known Browder fixed-point theorem. The stability of this scheme is established in L∞-norm. The uniqueness and convergence of the solution are analyzed. We discuss an iterative algorithm for solving the difference scheme and prove its convergence.

Transient dynamic response of fluid-saturated soil under a moving cyclic loading

Based on the Theory of Porous Media (TPM), a mathematical model of a two-dimensional incompressible fluid-saturated elastic soil is established, and the periodic boundary conditions are presented to analyze the transient dynamic response of this soil under a moving cyclic loading. The differential quadrature method (DQM) and the second-order backward difference scheme are applied to discretize the governing equations on the spatial and temporal domains, respectively. As application, a typical two-dimensional wave-induced transient problem with a seabed of finite thickness is analyzed, and the numerical results are compared with the analytical results presented in the present work. In addition, a transient dynamic response of fluid-saturated soil under limit moving vehicle loadings is studied. The effects of the velocity of vehicle and the volume fraction on the settlement and the pore water pressure are studied.

On first- and second-order difference schemes for differential-algebraic equations of index at most two

Difference schemes of the Euler and trapezoidal types for the numerical solution of the initial-value problem for linear differential-algebraic equations are examined. These schemes are analyzed for model examples, and their superiority over the familiar first- and second-order implicit methods is shown. Conditions for the convergence of the proposed algorithms are formulated.

Inverse Mode Problem in the Discrete Model of Circular Plate Axial Symmetry Vibration

Using the second-order center difference scheme, the difference discrete model of axial symmetry vibration of a circular plate with arbitrary supports was established. In this paper, the stiffness matrix of the discrete model is proved as a sign-oscillation matrix, obtaining the qualitative properties of the axial symmetry vibration of the system. Furthermore, the inverse mode problem was raised and solved. Three examples of the numerical computation were also given in the paper.

Simulation of radiation expansion of laser plasma in an external magnetic field

Radiation expansion of laser plasma in an external magnetic field is investigated in the paper. A two-dimensional system of ideal magneto-hydrodynamics with radiation transfer in a cylindrical system of coordinates was solved numerically using second-order conservative TVD difference scheme by space and time. A multigroup flux-limited diffusion scheme was applied for the solution of the radiative transfer equation. At the initial moment, the heating of a target, consisting of vapors of aluminum was implemented by a short-action laser pulse with a duration time of 30 nanoseconds and Gaussian profile by space with a half-thickness of 0.03 centimeters. Cases that take into account, as well as those that do not take into account, the radiation transfer and the magnetic field effects are considered. The numerical simulations show that inclusion of radiation transfer changes the dynamics of laser expansion quantitatively and qualitatively.

Application of factor difference scheme to solving discrete flow equations based on unstructured grid

A second-order mixing difference scheme with a limiting factor is deduced with the reconstruction gradient method and applied to discretizing the Navier-Stokes equation in an unstructured grid. The transform of nonorthogonal diffusion items generated by the scheme in discrete equations is provided. The Delaunay triangulation method is improved to generate the unstructured grid. The computing program based on the SIMPLE algorithm in an unstructured grid is compiled and used to solve the discrete equations of two types of incompressible viscous flow. The numerical simulation results of the laminar flow driven by lid in cavity and flow behind a cylinder are compared with the theoretical solution and experimental data respectively. In the former case, a good agreement is achieved in the main velocity and drag coefficient curve. In the latter case, the numerical structure and development of vortex under several Reynolds numbers match well with that of the experiment. It is indicated that the factor difference scheme is of higher accuracy, and feasible to be applied to Navier-Stokes equation.

Evolution of laser-produced plasma in an external magnetic field

The evolution of laser-produced plasma in an external magnetic field was studied. Equations of ideal MHD in a cylindrical coordinate system were solved numerically by using the conservative TVD difference scheme of second order in time and space. At the beginning, a short 30-ns laser pulse with the Gaussian transverse intensity distribution with a half-width of 0.03 cm was applied to heat the target that consisted of aluminum vapor plasma. The cases of weak (plasma parameter β = 1) and strong (β = 0.026) external magnetic fields were considered. The results of the numerical calculations show that the magnetic field increases the width of the laser plume’s front and forces plasma to move predominantly along the magnetic field lines. An increase of the magnetic field strength resulted in increased inhomogeneity of temperature and density distributions in the laser plume volume. The model shows that in the final stages of evolution the laser plasma takes the form of a confined jet aligned along the symmetry axis.

Navier–Stokes spectral solver in a sphere

The paper presents the first implementation of a primitive variable spectral method for calculating viscous flows inside a sphere. A variational formulation of the Navier–Stokes equations is adopted using a fractional-step time discretization with the classical second-order backward difference scheme combined with explicit extrapolation of the nonlinear term. The resulting scalar and vector elliptic equations are solved by means of the direct spectral solvers developed recently by the authors. The spectral matrices for radial operators are characterized by a minimal sparsity – diagonal stiffness and tridiagonal mass matrix. Closed-form expressions of their nonzero elements are provided here for the first time, showing that the condition number of the relevant matrices grows as the second power of the truncation order. A new spectral elliptic solver for the velocity unknown in spherical coordinates is also described that includes implicitly the Coriolis force in a rotating frame, but requires a minimal coupling between the modal velocity components in the Fourier space. The numerical tests confirm that the proposed method achieves spectral accuracy and ensures infinite differentiability to all orders of the numerical solution, by construction. These results indicate that the new primitive variable spectral solver is an effective alternative to the spectral method recently proposed by Kida and Nakayama, where the velocity field is represented in terms of poloidal and toroidal functions.

Modified stability controllable second-order difference scheme for incompressible flow and heat transfer problems

This paper mainly investigates the Stability Controllable Second-order Difference (SCSD) scheme for solving 2D convection-diffusion problems, a strong implicit iteration procedure is adopted together with the block correction technology and Pent-Diagonal Matrix Algorithm (PDMA). Non-uniform grid technology is also used for treating the nodes close to the domain boundaries. The scheme is verified via two benchmark problems, i.e., the lid-driven cavity flow and the natural convection in a cavity. It has been shown that the scheme has a rather good calculation stability and accuracy, implying that the proposed method has a wide applicability.

DQM for dynamic response of fluid-saturated visco-elastic porous media

Based on the Porous Media Theory (PMT), a mathematical model of space-axisymmetrical problems for incompressible fluid-saturated visco-elastic porous media is presented in the case of small deformation, in which the differential-type constitutive relation is applied to describe the mechanical characteristics of solid skeleton. The differential quadrature method (DQM) and the second-order backward difference scheme are used to discretize the governing equations on the spatial and temporal domains, respectively, and a method is proposed to deal with the singularity conditions at points located on the symmetry axis. As application, the dynamic behavior of a column of fluid-saturated elastic porous media is analyzed firstly. The obtained results are compared with the analytical results in the existing literature, they are comparatively accordant, which means that the model and method presented in this paper are correct, and the obtained results are reliable. Further, the dynamic response of a space-axisymmetrical body of fluid-saturated visco-elastic porous media is analyzed, in which the material characteristic of the solid skeleton is described by Burgers model with four parameters.

A second-order accurate non-linear difference scheme for the N -component Cahn–Hilliard system

We consider a second-order conservative nonlinear numerical scheme for the N -component Cahn–Hilliard system modeling the phase separation of a N -component mixture. The scheme is based on a Crank–Nicolson finite-difference method and is solved by an efficient and accurate nonlinear multigrid method. We numerically demonstrate the second-order accuracy of the numerical scheme. We observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results. We also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture. We investigate the effects of a concentration dependent mobility on phase separation.

Computational properties of vertical grids for a nonhydrostatic anelastic model

Computational dispersion properties of all vertically staggered grids, which are presently available, are analysed in terms of frequency and group velocity components using the second-order centre difference scheme for a nonhydrostatic anelastic approximation system with a general method. The inertial-gravitational waves with a horizontal scale of a hundred-, ten- and one-kilometres are considered. The comparison analysis shows that the Charny-Phillips (CP) and Lorenz grids are suitable for waves at all abovementioned horizontal scales, while the Lorenz time staggered and Charny-Phillips time staggered grids are applicable only to waves with a horizontal scale less than 10 km. The unstaggered (N) grid is not suitable for simulating waves at any horizontal scale. In an idealised flow numerical test, the result on the CP grid has much less error than that on the N grid.

A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.

Design of High-Order Difference Scheme and Analysis of Solution Characteristics—Part I: General Formulation of High-Order Difference Schemes and Analysis of Convective Stability

In this article, a general design method of second-order difference schemes is presented. By this method, we can easily design any second-order or higher-order difference scheme instead of using complex Lagrange interpolation methods or spline functions. Moreover, it is proved that all existing second-order difference schemes in numerical heat transfer fit this general design style. In addition, based on this general style of second-order scheme, the general style of a second-order absolutely stable scheme is deduced, and the stability definitions are shown in a normalized variable diagram. Finally, through studying the solution characteristics of 14 second-order difference schemes, it is found that, to second order precise, absolutely stable schemes obtained from the general method can achieve good convergence even when the grid Pelect number reaches 100,000. However, at the same time, the false diffusion of the scheme tends to increase along with the increasing value of a i (coefficient in the interface variable definition).

Design of High-Order Difference Scheme and Analysis of Solution Characteristics—Part II: A Kind of Third-Order Difference Scheme and New Scheme Design Theory

In this Part II, on the basis of the general style design of second-order difference scheme and the analysis of the absolutely stable scheme proposed in Part I, the companion article, the general design method of any high-order difference scheme is proposed. Based on this method, a new kind of third-order difference scheme including 17 different variants is constructed, which uses the same grid points as existing second-order difference schemes but is different from them in that the grids are chosen symmetrically from two sides of the interface. Because they have the same matrix style created by the same grid plots of the discretization equation, these third-order schemes require the same CPU time and memory as the second-order schemes; however, this kind of symmetrical third-order difference scheme will keep the consistency between the false diffusion and the stability, and the stability of the scheme is better than that of the existing biased second-order scheme. Further research shows that under the conditions of matrix style and computer memory, the scheme constituted by symmetrically numbered grids from two sides of the interface with odd order of accuracy can maintain consistency between numerical accuracy and stability better than any kind of scheme designed according to the “upwind” idea. Based on this understanding, a new scheme design theory called symmetric and odd-order accuracy scheme design theory is proposed.

On the Numerical Stability of Some Symplectic Integrators

In this paper, we analyze the linear stabilities of several symplectic integrators, such as the first-order implicit Euler scheme, the second-order implicit mid-point Euler difference scheme, the first-order explicit Euler scheme, the second-order explicit leapfrog scheme and some of their combinations. For a linear Hamiltonian system, we find the stable regions of each scheme by theoretical analysis and check them by numerical tests. When the Hamiltonian is real symmetric quadratic, a diagonalizing by a similar transformation is suggested so that the theoretical analysis of the linear stability of the numerical method would be simplified. A Hamiltonian may be separated into a main part and a perturbation, or it may be spontaneously separated into kinetic and potential energy parts, but the former separation generally is much more charming because it has a much larger maximum step size for the symplectic being stable, no matter this Hamiltonian is linear or nonlinear.

An improved numerical scheme to compute horizontal gradients at the ice-sheet margin: its effect on the simulated ice thickness and temperature

AbstractIn three-dimensional numerical ice-sheet models that use finite-difference schemes, the position of ice margins is poorly represented with a regular quadratic grid. As a result, in a centered difference scheme, the surface gradient term and the flux divergence term computed for the gridpoints next to the ice margin may be inaccurate. In this paper, an improved scheme is presented that computes the horizontal gradients at the ice-sheet margin using an asymmetric (upstream) second-order difference scheme in order to avoid using information from the zero-thickness gridpoints. The model is applied to an idealized synthetic geometry to obtain a steady-state ice-sheet topography. The improved model shows a realistically smooth thickness distribution near the margin. Thermomechanical coupling is found to enhance the error near the margin. The error in simulated thicknesses with the centered-difference method was significantly reduced with the new upstream scheme.

First- and second-order accurate implicit difference schemes for the numerical resolution of the generalized Charney–Obukhov and Hasegawa–Mima equations

First- and second-order accurate implicit difference schemes for the numerical solution of the nonlinear generalized Charney-Obukhov and Hasegawa-Mima equations with scalar nonlinearity are constructed. On the basis of numerical calculations accomplished by means of these schemes, the dynamics of two-dimensional nonlinear solitary vortical structures are studied. The problem of stability for the first-order accurate semi-discrete scheme is investigated. The dynamic relation between solutions of the generalized Charney-Obukhov and Hasegawa Mima equations is established. It is shown that, contrary to existing opinion, the scalar nonlinearity in the ease of the generalized Hasegawa-Mima equation develops monopolar anticyclone, while in case of the generalized Charnev -Obukhov equation it develops monopolar cyclone.

Numerical Simulation of Transport in Optical Fiber Drawing with Core–Cladding Structure

Optical fibers are typically drawn from silica preforms, which usually consist of two concentric cylinders called the core and the cladding, heated in a high-temperature furnace. For optical communication purposes, the core always has a higher refractive index than the cladding to obtain total internal reflection. In order to investigate the effect of this core–cladding structure on optical fiber drawing, a numerical model has been developed in this work. Axisymmetric flows of a double-layer glass and aiding purge gas in a concentric cylindrical furnace are considered. The thermal and momentum transport in both glass layers and gas are coupled at the interface boundaries. The neck-down profile is generated using an iterative numerical scheme. The zonal method is applied to model the radiation transfer in the glass preform. The gas is taken as nonparticipating. Coordinate transformations are used to convert the resulting complex domains into cylindrical regions. The stream function, vorticity, and energy equations for the core, the cladding, and the purge gas are solved by finite difference methods, using a false transient approach coupled with the alternating direction implicit method. A second-order differencing scheme is used for discretization. The numerical results are validated by comparing with results available in the literature. The effects of changes in the refractive index and absorption coefficient due to doping on fiber drawing are investigated. This problem has received very little attention in the literature, particularly with respect to modeling, and this paper presents an initial study of the underlying transport.