A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.     

求解非线性刚性初值问题的隐显线性多步方法的误差分析

隐显线性多步方法由隐式线性多步方法和显式线性多步法组合而成.本文主要讨论求解满足单边Lipschitz条件的非线性刚性初值问题和一类奇异摄动初值问题的隐显线性多步方法的误差分析.最后,由数值例子验证了所获的理论结果的正确性及方法处理这两类问题的有效性.     

一类非线性方程组的Newton-PSS迭代法

正定反Hermite分裂(PSS)方法是求解大型稀疏非Hermite正定线性代数方程组的一类无条件收敛的迭代算法.将其作为不精确Newton方法的内迭代求解器,我们构造了一类用于求解大型稀疏且具有非Hermite正定Jacobi矩阵的非线性方程组的不精确Newton-PSS方法,并对方法的局部收敛性和半局部收敛性进行了详细的分析.数值结果验证了该方法的可行性与有效性.     

Linearly implicit methods for nonlinear parabolic equations

We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes are locally stable. We establish optimal order error estimates.

     

一类弱非线性方程组的Picard-MHSS迭代方法

修正的Hermite/反Hermite分裂(MHSS)迭代方法是一类求解大型稀疏复对称线性代数方程组的无条件收敛的迭代算法.基于非线性代数方程组的特殊结构和性质,我们选取Picard迭代为外迭代方法,MHSS迭代作为内迭代方法,构造了求解大型稀疏弱非线性代数方程组的Picard-MHSS和非线性MHSS-like方法.这两类方法的优点是不需要在每次迭代时均精确计算和存储Jacobi矩阵,仅需要在迭代过程中求解两个常系数实对称正定子线性方程组.除此之外,在一定条件下,给出了两类方法的局部收敛性定理.数值结果证明了这两类方法是可行、有效和稳健的.     

求解带非线性边界条件的反应扩散方程数值解的组合迭代方法

1　引　　言我们首先考虑如下抛物型方程ｕｔ-ＤΔｕ =ｆ(ｘ ,ｔ ,ｕ) (ｔ∈ ( 0 ,Ｔ],ｘ∈Ω ) ｕ/ ν+ βｕ =ｇ(ｘ ,ｔ ,ｕ) (ｔ∈ ( 0 ,Ｔ],ｘ∈ Ω )ｕ(ｘ ,0 ) =ψ(ｘ) (ｘ∈Ω )( 1 .1 )其中Ｔ为正常数 ,Ω 是ＲＰ 空间的有界区域 记ＱＴ=Ω × ( 0 ,Ｔ],ＳＴ= Ω × ( 0 ,Ｔ],假设在ＱＴ上Ｄ≡ｄ(ｘ ,ｔ) >0 ,在ＳＴ 上β≡β(ｘ ,ｔ)≥ 0 又设 ｆ(ｘ ,ｔ,ｕ) ,ｇ(ｘ ,ｔ,ｕ)为关于ｕ的非线性函数 ,且对ｘ ,ｔ各参数满足Ｈ¨ｏｌｄｅｒ连续条件 将 ( 1 .1 )离散化之后我们得到相应的有限差分系统 ,当 ｇ(ｘ ,ｔ,ｕ)为ｕ的线性…     

Waveform relaxation methods for implicit differential equations

We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying Runge-Kutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initial-value problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.     

Multiple right-hand side techniques for the numerical simulation of quasistatic electric and magnetic fields

The simulation of slowly varying transient electric high-voltage fields and magnetic fields requires the repeated and successive solution of high-dimensional linear algebraic systems of equations with identical or near-identical system matrices and different right-hand side vectors. For these solution processes which are required within implicit time integration schemes and nonlinear (quasi-)Newton–Raphson methods an iterative multiple right-hand side (mrhs) scheme is used which recycles vector subspaces resulting from previous preconditioned conjugate gradient iteration runs. The combination of this scheme with a subspace projection extrapolation start value generation scheme is discussed. Numerical results for three-dimensional electric and magnetic field simulations are presented and the efficiency of the new schemes re-using eigenvector information from previous iteration processes with different tolerance criteria are compared to those of standard conjugate gradient iterations.     

IMEX method convergence for a parabolic equation

Although implicit-explicit (IMEX) methods for approximating solutions to semilinear parabolic equations are relatively standard, most recent works examine the case of a fully discretized model. We show that by discretizing time only, one can obtain an elementary convergence result for an implicit-explicit method. This convergence result is strong enough to imply existence and uniqueness of solutions to a class of semilinear parabolic equations.     

IMPLICIT-EXPLICIT RUNGE-KUTTA-ROSENBROCK METHODS WITH ERROR ANALYSIS FOR NONLINEAR STIFF DIFFERENTIAL EQUATIONS

Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear stiff ordinary differential equations by combining linearly implicit Rosenbrock methods with ex-plicit Runge-Kutta methods.First,the general order conditions up to order 3 are obtained.Then,for the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condi-tion and a class of singularly perturbed initial-value problems,the corresponding errors of the implicit-explicit methods are analysed.At last,some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the methods.     

Split Newton iterative algorithm and its application

Inspired by some implicit-explicit linear multistep schemes and additive Runge-Kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Consistency and global convergence of the new method are also maintained. To test its effectiveness, we apply the method to nonlinear reaction-diffusion equations, such as Burger’s-Huxley equation and fisher’s equation. Numerical examples suggest that the involved iterative method is much faster than the classical Newton’s method on a given time interval.     

A numerical study of air-vapor-heat transport through textile materials with a moving interface

This paper focuses on the numerical study of heat and moisture transfer in clothing assemblies, which is described by a multi-component and multiphase air-vapor-heat flow with a moving interface. A splitting semi-implicit finite volume method is applied for the system of nonlinear parabolic equations and an implicit Euler scheme is used for the interface equation. In terms of classical Dirichlet to Neumann map, the implicit system can be solved directly and no iteration is needed. Two types of clothing assemblies are investigated and the comparison with experimental measurements is also presented.     

离散空间分数阶非线性薛定谔方程的MHSS型迭代方法

利用隐式守恒型差分格式来离散空间分数阶非线性薛定谔方程,可得到一个离散线性方程组.该离散线性方程组的系数矩阵为一个纯虚数复标量矩阵、一个对角矩阵与一个对称Toeplitz矩阵之和.基于此,本文提出了用一种\textit{修正的埃尔米特和反埃尔米特分裂}(MHSS)型迭代方法来求解此离散线性方程组.理论分析表明,MHSS型迭代方法是无条件收敛的.数值实验也说明了该方法是可行且有效的.     

A numerical study of air–vapor–heat transport through textile materials with a moving interface

This paper focuses on the numerical study of heat and moisture transfer in clothing assemblies, which is described by a multi-component and multiphase air–vapor–heat flow with a moving interface. A splitting semi-implicit finite volume method is applied for the system of nonlinear parabolic equations and an implicit Euler scheme is used for the interface equation. In terms of classical Dirichlet to Neumann map, the implicit system can be solved directly and no iteration is needed. Two types of clothing assemblies are investigated and the comparison with experimental measurements is also presented.     

High-order methods for parabolic problems

Error estimates valid for all t ? 0 for the semi-discrete Galerkin approximation of a parabolic mixed boundary-initial value problem are presented. The solution of the resulting system of ordinary differential equations by implicit Runge-Kutta formulae for arbitrarily high order of accuracy, are discussed. Strongly A-stable methods are found to be advantageous. Theoretical and experimental results for the solution of the resulting system of algebraic equations using a preconditioned outer iteration scheme are discussed. Even the inner linear algebraic equations are preferably solved by iteration.     

非线性波动方程的弱隐式与显式差分方法

广泛出现于物理、化学、机械动力学、生物、几何学等领域的非线性波动方程已经有很多的研究工作,Sine-Gordon方程和非线性受迫振动方程就是典型的例子.周毓麟教授在[1]中研究了非线性波动方程组     

一种基于Chebyshev迭代解非线性方程组的方法

本文根据牛顿迭代和Chebyshev迭代法给出了一种新的迭代方法,该方法有较高的收敛阶,并在理论上给予了证明.最后给出了四个实例,将本文的实验结果与现有的几种方法的实验结果进行比较,表明我们的方法迭代次数少,有明显的优势.     

求非线性方程组和优化问题全部解的胞腔方法

求非线性方程组和优化问题全部解的胞腔方法张讲社，游兆永，徐宗本（西安交通大学数学系）ＡＣＥＬＬＭＥＴＨＯＤＦＯＲＦＩＮＤＩＮＧＡＬＬＳＯＬＵＴＩＯＮＳＯＦＮＯＮＬＩＮＥＡＲＥＱＵＡＴＩＯＮＳＡＮＤＯＰＴＩＭＩＺＡＴＩＯＮＰＲＯＢＬＥＭＳ￥ＺｈａｎｇＪ...     

一类非线性系统的分段迭代学习控制

本文提出并研究一类非线性系统的分段迭代学习控制问题.基于P型学习律和D型学习律构建得到分段迭代学习控制律,利用压缩映射原理,证明这种分段迭代学习律能使得系统的输出跟踪误差沿迭代轴方向收敛.仿真算例验证了算法的有效性.     

Modified Implicit–Explicit BDF Methods for Nonlinear Parabolic Equations

Implicit–explicit multistep methods for nonlinear parabolic equations were recently analyzed. If the implicit scheme is one of the backward differentiation formulae (BDF) of order up to six, then the corresponding implicit–explicit method of the same order is stable provided the stability constant is less than a specific scheme-dependent constant. Based on BDF, implicit methods are constructed such that the corresponding implicit–explicit scheme of the same order exhibits improved stability properties.