Expanding stability regions of explicit advective-diffusive finite difference methods by Jacobi preconditioning

Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel. The disadvantage of explicit methods is the severe restrictions that are placed on stable time-step intervals. Stability bounds for explicit time differencing methods on advective–diffusive problems are generally determined by the diffusive part of the problem. These bounds are very small and implicit methods are used instead. The linear systems arising from these implicit methods are generally solved by iterative methods. In this article we develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods. A Jacobi preconditioned time differencing method is defined and analyzed for both diffusion and advection–diffusion equations. Several computational examples of both linear and nonlinear advective-diffusive problems are solved to demonstrate the accuracy and improved stability limits. © 1995 John Wiley & Sons, Inc.     

Optimized composite finite difference schemes for atmospheric flow modeling

In this paper, we use some finite difference methods in order to solve an atmospheric flow problem described by an advection–diffusion equation. This flow problem was solved by Clancy using forward‐time central space (FTCS) scheme and is challenging to simulate due to large errors in phase and amplitude which are generated especially over long propagation times. Clancy also derived stability limits for FTCS scheme. We use Von Neumann stability analysis and the approach of Hindmarsch et al. which is an improved technique over that of Clancy in order to obtain the region of stability of some methods such as FTCS, Lax–Wendroff (LW), Crank–Nicolson. We also construct a nonstandard finite difference (NSFD) scheme. Properties like stability and consistency are studied. To improve the results due to significant numerical dispersion or numerical dissipation, we derive a new composite scheme consisting of three applications of LW followed by one application of NSFD. The latter acts like a filter to remove the dispersive oscillations from LW. We further improve the composite scheme by computing the optimal temporal step size at a given spatial step size using two techniques namely; by minimizing the square of dispersion error and by minimizing the sum of squares of dispersion and dissipation errors.     

关于色散方程的一类二阶恒稳显格式

1　引　　言对于具有高阶空间导数的发展方程 ,其显格式因结构简单 ,易于计算 ,具有明显的计算优越性 ,但已有的绝大多数显格式的稳定性条件都十分苛刻 (见 [6 ] -[1 5] ) ,远不如一般隐格式 ,使其应用受到限制 .1 994年《计算物理》中关于“色散方程的一类具任意稳定性的显格式”一文 (见 [1 4 ] ) ,把色散方程显格式的稳定性条件提高到了可以任意选择的程度 ,但截断误差仅为 O(τ+h) .本文构造了新一类双参数显式差分格式 ,它是绝对稳定的 ,且其截断误差是 O(τ+h2 ) ,它结构简单 ,易于实现计算 ,利于实际应用 .我们用数值例子验证了理论…     

一维气液两相漂移模型全隐式AUSMV算法研究

气液两相漂移模型显式AUSMV（advection upstream splitting method combined with flux vector splitting method）算法的时间步长受限于CFL（Courant-Friedrichs-Lewy）条件，为了提高计算效率，建立了一种全隐式AUSMV算法求解气液两相漂移模型.采用AUSM格式结合FVS（flux vector splitting）格式构造连续方程和运动方程的对流项数值通量，AUSM格式构造压力项数值通量.离散控制方程是非线性方程组，采用六阶Newton(牛顿)法结合数值Jacobi矩阵求解.计算经典算例Zuber-Findlay激波管问题和复杂漂移关系变质量流动问题，结果分析表明:全隐式AUSMV算法，色散效应小，无数值震荡，计算精度高.在压力波波速高的条件下，可以显著提高计算效率，耗散效应小.     

Partitioned time discretization for parallel solution of coupled ODE systems

One decoupling method for multiphysics, multiscale, multidomain applications involves partitioning the problem via explicit time discretizations in the coupling terms. Specialized, problem-specific techniques are needed for the resulting partitioned methods to avoid time step restrictions which make long time calculations costly. This report studies unconditionally stable, uncoupled time stepping methods for a model problem sharing mathematical structure akin to the coupled atmosphere-ocean system. Three decoupled time stepping algorithms are introduced and their stability and consistency are rigorously examined. Numerical experiments are performed that study their stability and convergence properties.     

完整Coriolis力作用下的非线性近惯性波

从包含有完整Coriolis力作用下的大气运动原始基本方程组出发,通过尺度分析,采用多重尺度法及摄动展开法,推导了中高纬大气非线性近惯性波振幅演化所满足的Korteweg-de Vries方程.从演化方程的结果可以看出Coriolis参数水平分量对非线性近惯性波的影响,主要体现为对频散效应的修正及与基本流的相互作用.从理论上解释了完整Coriolis力作用下的中高纬地区大气非线性近惯性波运动的物理机制.     

Wavelet-Taylor Galerkin Method for the Burgers Equation

In this paper, we propose a wavelet-Taylor Galerkin method for the numerical solution of the Burgers equation. In deriving the computational scheme, Taylor-generalized Euler time discretization is performed prior to wavelet-based Galerkin spatial approximation. The linear system of equations obtained in the process are solved by approximate-factorization-based simple explicit schemes, and the resulting solution is compared with that from regular methods. To deal with transient advection-diffusion situations that evolve toward a convective steady state, a splitting-up strategy is known to be very effective. So the Burgers equation is also solved by a splitting-up method using a wavelet-Taylor Galerkin approach. Here, the advection and diffusion terms in the Burgers equation are separated, and the solution is computed in two phases by appropriate wavelet-Taylor Galerkin schemes. Asymptotic stability of all the proposed schemes is verified, and the L_∞ errors relative to the analytical solution together with the numerical solution are reported. AMS subject classification (2000) 65M70     

Improved stability techniques for the solution of Navier-Stokes equations

When solving the Navier-Stokes equations for transient incompressible viscous flow problems, one normally encounters a decrease in numerical stability of the time integration algorithm with an increase in Reynolds number. This instability cannot be easily overcome due to the non-linearities present. The present paper, using the finite element method to integrate the equations in the spacial dimension, incorporates a time-staggered semi-implicit fractional step technique to improve stability at the higher Reynolds numbers. Unlike the upwind or directional differencing schemes normally employed to increase numerical stability, the present scheme does not introduce numerical damping or artificial viscocity, and becomes more stable as the Reynolds number increases. Results for this scheme are compared with various explicit integration schemes for the case of flow around a circular cylinder at Reynolds numbers of 100 to 400. For comparable accuracy the time-staggered semi-implicit fractional step technique was found to be up to 25 times more efficient than the other explicit integration schemes.     

二维抛物型方程简单实用的显格式

提出了一个解二维抛的型方程初边值问题的简单实用的显格式，证明了其截断误差阶是Ｏ，稳定性条件是α＋β≠１／２且ｍａｘ｛α，β｝≤１／４，其中，α＝α．Δｔ／Δｘ＾２．β＝α．Δｔ／Δｙ＾２。     

解三维热传导方程的一种高精度的显格式

对解三维热传导方程利用待定参数方法构造出一种精度O(Δt2+Δx4+Δy4+Δz4)的高精度易于计算的显式差分格式,并给出了其稳定性,通过数值例子可见其精度较其它方法提高2～3位有效数字。     

Mathematical study of two-variable systems with adaptive numerical methods

In this paper, we consider reaction–diffusion systems arising from two-component predator–prey models with Smith growth functional response. The mathematical approach used here is in two folds since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with the fourth-order difference operator and advance the resulting nonlinear system of ordinary differential equations with the two competing families of the exponential time differencing (ETD) schemes, and we analyze them for stability. Numerical comparison between these two methods for solving various predator–prey population models with functional responses are also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled.     

On Meeting Energy Balance Errors in Co-Simulations

The term co-simulation denotes the coupling of some simulation tools for dynamical systems into one big system by having them exchange data at points of a fixed time grid and extrapolating the received data into the interval, while none of the steps is repeated for iteration. From the global perspective, the simulation thus has a strong explicit component. Frequently, among the data passed across subsystem boundaries there are flows of conserved quantities, and as there is no iteration of steps, system-wide balances may not be fulfilled: the system is not solved as one monolithic equation system. If these balance errors accumulate, simulation results become inaccurate. Balance correction methods which compensate these errors by adding corrections for the balances to the signal in the next coupling time step have been considered in past research. But establishing the balance of one quantity a posteriori due to the time delay in general cannot establish the balances of quantities that depend on the exchanged quantities, usually energy. In most applications from physics, the balance of energy is equivalent to stability. In this paper, a method is presented which allows users to choose the quantity that should be balanced to be that energy, and to accurately balance it. This establishes also numerical stability for many classes of stable problems.     

Peculiarities of error accumulation in solving problems for simple equations of mathematical physics by finite difference methods

A mixed problem for the one-dimensional heat conduction equation with several versions of initial and boundary conditions is considered. To solve this problem, explicit and implicit schemes are used. Sweep and iterative methods are used for the implicit scheme when solving the system of equations. Numerical filtering of a finite sequence of results obtained for various grids with an increasing number of node points is used to analyze the method and rounding errors. To investigate rounding errors, the results obtained for various machine word mantissa lengths are compared. The numerical solution of a mixed problem for the wave equation is studied by similar methods. Some deterministic dependencies of the numerical method and rounding errors on the spatial coordinates, time, and the number of nodes are found. Some models of sources to describe the behavior of the errors in time are constructed. They are based on the results of computational experiments under various conditions. According to these models, which have been experimentally verified, the errors increase, decrease, or stabilize in time under the conditions, similarly to energy or mass.     

Stability Analysis and Application of the Exponential Time Differencing Schemes

Exponential time differencing schemes are time integration methods that can be efficiently combined with spatial spectral approximations to provide very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. We study in this paper the stability properties of some exponential time differencing schemes. We also present their application to the numerical solution of the scalar Allen-Cahn equation in two and three dimensional spaces.     

瞬态问题稳定性及弥散分析的无网格RKPM配点法研究

介绍了基于强形式的RKPM配点法求解瞬态动力问题的算法,并提出了采用RKPM配点法,配合时间域中心差分求解二阶波动方程的稳定性评价方法,并通过数值算例验证了此方法的正确性．此评价方法可以方便有效地评估出实际计算时的临界时间步长．通过数值算例比较可知,实际算例的计算临界时间步长与本评价方法,所预测的临界时间步长结果非常接近．给出了如何合理地选择RKPM形函数支撑域的建议．最后与径向基函数配点法进行了对比研究．     

Convergence of a finite volume scheme for coagulation-fragmentation equations

This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation. A time explicit finite volume scheme is developed, based on a conservative formulation of the equation. It is shown to converge under a stability condition on the time step, while a first order rate of convergence is established and an explicit error estimate is given. Finally, several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution.

     

Dispersion and stability properties of radial basis collocation method for elastodynamics

Strong form collocation with radial basis approximation, called the radial basis collocation method (RBCM), is introduced for the numerical solution of elastodynamics. In this work, the proper weights for the boundary collocation equations to achieve the optimal convergence in elastodynamics are first derived. The von Neumann method is then introduced to investigate the dispersion characteristics of the semidiscrete RBCM equation. Very small dispersion error (< 1%) in RBCM can be achieved compared to linear and quadratic finite elements. The stability conditions of the RBCM spatial discretization in conjunction with the central difference temporal discretization are also derived. We show that the shape parameter of the radial basis functions not only has strong influence on the dispersion errors, it also has profound influence on temporal stability conditions in the case of lumped mass. Further, our stability analysis shows that, in general, a larger critical time step can be used in RBCM with central difference temporal discretization than that for finite elements with the same temporal discretization. Our analysis also suggests that although RBCM with lumped mass allows a much larger critical time step than that of RBCM with consistent mass, the later offers considerably better accuracy and should be considered in the transient analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Quadratic spline collocation for one-dimensional linear parabolic partial differential equations

New methods for solving general linear parabolic partial differential equations (PDEs) in one space dimension are developed. The methods combine quadratic-spline collocation for the space discretization and classical finite differences, such as Crank-Nicolson, for the time discretization. The main computational requirements of the most efficient method are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth order. The stability and convergence properties of some of the new methods are analyzed for a model problem. Numerical results demonstrate the stability and accuracy of the methods. Adaptive mesh techniques are introduced in the space dimension, and the resulting method is applied to the American put option pricing problem, giving very competitive results.     

An asymptotic solution to non-linear transmission lines

This paper explores an asymptotic approach to the solution of a non-linear transmission line model. The model is based on a set of non-linear partial differential equations without analytical solution. The perturbations method is used to reduce the system of non-linear equations to a single non-linear partial differential equation, the modified Korteweg–de Vries equation (KdV). By using the Laplace transform, the solution is represented in integral form in terms of Green's functions. The solution for the non-linear case is obtained by means of asymptotic methods. Thus, an approximate explicit analytical solution to the problem is obtained where the errors can be controlled. This allows us to analyze the non-linear behavior of the solution. This kind of information is difficult to obtain by means of numerical methods due to the fact that for large periods of time greater computational resources are required and also accumulated errors increase. For this reason, asymptotic methods have a great importance like a natural complement to numerical methods. Computer simulations support the developments presented.     

On a problem of correction of the controlled process : PMM vol. 42, no. 6, 1978, pp. 1127–1131

The results obtained in [1] are developed further, by considering a case in which the processs is described by a nonlinear, differential vector equation of the p-th order, the correction vector appears in the right hand side of the equation of motion as one of the terms, and the noise vector is a measurable vector function. A constructive method of correcting the process in question is given, with the correction equation obtained in the explicit form.