一类解Stiff常微分方程组初值问题的多级隐式Hybrid方法

§1.引言 为了叙述上的简洁而又不失一般性,我们考虑Stiff常微分方程组自治系统初值问题 y′-f(y),y(0)=Y_0 (1.1)的数值解,在此假定F(y)有适当的可微性并用y(z)表示(1.1)的精确解,用y_n表示(1.1)在x=nh点的数值解。h为积分步长,记f_n=f(y_n)。 在[1]中,Dahlquist证明了A稳定的线性多步法所能达到的最高阶是2。1978年Wanner等人进一步证明了A稳定的线性方法所能达到的最高阶不能超过2q,其中q可以是多导方法的最高导数或者是Runge-Kutta方法的级数。也就是说,高阶A稳定的方法只出现在像隐式多导方法或隐式多级Runge-Kutta方法等一类方法中。本文暂只涉及     

Birkhoff意义下Hojman Urrutia方程的离散变分计算

在Birkhoff框架下,采用离散变分方法研究了非Hamilton系统-Hojman-Urrutia方程的数值解法,并通过和传统的Runge-Kutta方法进行比较,说明了在Birkhoff框架下研究这类不具有简单辛结构的非Hamilton系统可以得到更可靠和精确的数值结果.     

Birkhoff框架下Whittaker方程的离散变分算法

本文在Birkhoff框架下,采用离散变分方法研究了非Hamilton系统-Whittaker方程的数值解法,并通过和传统的Runge-Kutta方法进行比较,说明了在Birkhoff框架下研究非Hamilton系统可以得到更加可靠和精确的数值结果.     

分数阶细胞神经网络自适应同步控制设计及电路仿真

构建一个新的分数阶细胞神经网络系统,设计驱动系统非线性参数已知而响应系统非线性参数值未知的驱动–响应系统,运用自适应同步控制器及参数自适应调整律实现该驱动–响应系统同步.数值仿真和动力学分析结果表明新的分数阶细胞神经网络系统具有混沌特性.结合分数阶电路理论设计出新的分数阶细胞神经网络系统同步控制的电路原理图.本方案实际可实现4096种多元组合电路,为简洁起见,选取分数阶qi(i=1,2,3)相同值(即q1=q2=q3=0.95)的组合电路进行电路仿真.仿真结果表明,多元电路仿真和数值仿真实验结果具有很高的吻合度.从而证实了该自适应同步控制方法在物理上的可实现性,在工程领域中具有现实的应用价值.     

机器人动力学与控制——讲座连载之四

§4 李雅普诺夫方法 RM系统的动能为 T=1/2q~TD(q)q (4.1)记广义冲量为p p=(T/q)~T=D(q)q (4.2) 式(4.1)、(4.2)中的q为RM系统的广义坐标q=[q_1,…,q_n]~T,设系统的势能为V(q),H=T+V(q)为系统的哈密顿函数,则RM动力学方程为     

关于拓展的向后微分公式的一点注记

1.引 言 我们考虑Stiff常微分方程初值问题 y'=f(x,y),a≤x≤b, (1.1) y(x_0)=y_0的数值解.在这里,以y(x)表示(1.1)的精确解,用y_n表示(1.1)在x=nh点的数值解,f_n=f(x_n,y_n)。 在中,Cash导出一类拓展的向后微分公式(以后称为Cash方法),其优点是它的绝对稳定区域比相应的向后微分公式(Gear方法)的绝对稳定区域大,方法的阶为p=     

浅水动边界问题的位移法模拟

为精确模拟浅水波非线性演化过程中的动边界,提出一种基于位移的Hamilton变分原理,并进而导出一种基于位移的浅水方程（Shallow Water Equation based on Displacement,SWE D）.SWE D以位移为基本未知量,可以精确满足动边界处的零水深要求并精确捕捉动态边界位置,且解具有协调性.在Hamilton变分原理的框架下,分别采用有限元和保辛积分算法对该浅水方程进行空间离散和时间积分,可有效地处理不平水底情况,保证对非线性演化进行长时间仿真的精度.数值算例表明该方法适用于浅水动边界问题的数值模拟.     

不含｛M（p，q），C3，C4｝作为导出子图的图的色数

Gyárfás 曾猜想,设 F是一个森林,对于每一个 F‐free 的图 G ,存在整数函数 f（F ,ω（G））使得χ（G）尘 f（F ,ω（G））。利用一个引理,得到了每一个不含｛M（p ,q）,C3,C4｝作为导出子图的图是（p ＋ q －1）‐可着色的。     

关于《一类非线性时滞系统的稳定化控制器设计研究》一定理的讨论

文献[1]将精确线性化的方法应用到一类时滞非线性系统稳定化控制器设计中,提出一种新的稳定化控制器设计方法,其思路是可取的.但是我认为其中的定理1有误,下面提出一家之言与作者商讨.考虑单输入非线性时滞系统x=f(x) g(x)u(t-τ),(1)其中x∈Rn,u∈R,f(.),g(.)为C∞非线性向量场;τ为时滞,f(0)=0.同时引入线性时滞系统w=Aw bu(t-τ),(2)其中w=[w1…wn]T∈Rn为新的状态变量A=01…0?0……0,　b=0?1.(3)　　原定理为定理1.对于非线性时滞系统(1),通过微分同胚变换w=T(x)将其转化成线性时滞系统(3)的充分必要条件是(i)rankM(x)=n,其中M(…     

关于QL-、D-蕴涵的分配性方程的解

研究了[r→（t∧s）]≡[（r→t）∧（r→s）],[r→（t∨s）]≡[（r→t）∨（r→s）],[（p∧q）→r]≡[（p→r）∨（q→r）],[（p∨q）→r]≡[（p→r）∧（q→r）]4个分配性方程,它们在模糊集理论中的形式分别是I（r,T₁（t,s））=T₂（I（r,t）,I（r,s））,I（r,S₁（t,s））=S₂（I（r,t）,I（r,s））,I（T₁（p,q）,r）=S₁（I（p,r）,I（q,r））,I（S₁（p,q）,r）=T₁（I（p,r）,I（q,r））,其中p,q,r,s,t∈[0,1],T₁、T₂为任意三角模,S₁、S₂为任意三角余模,给出了I为QL-、D-蕴涵时满足分配性方程的充要条件。     

显式和对角隐式Rung-Kutta方法求解中立型泛函微分方程的非线性稳定性

本文致力于研究巴拿赫空间中非线性中立型泛函微分方程显式和对角隐式Rung-Kutta方法的稳定性．获得了一些显式和对角隐式Rung-Kutta方法求解非线性中立型泛函微分方程的数值稳定性和条件收缩性结果,数值试验验证了这些结果．     

Construction and implementation of general linear methods for ordinary differential equations: A review

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge-Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge-Kutta stability is also briefly discussed.     

Efficient implementation of diagonally implicit Runge-Kutta methods

Efficient schemes for the implementation of diagonally implicit Runge-Kutta methods are considered. Methods of the 3rd and 4th orders are implemented. They are compared with known implicit solvers as applied to the solution of stiff and differential-algebraic equations.     

Stability of ROW methods for non autonomous systems of ordinary differential equations

AN-stability of ROW methods is studied. The concept of LN-equivalent schemes belonging to different classes of one-step methods is introduced to do it. Ways to construct ROW methods with improved stability for linear nonautonomous and nonlinear problems are studied using the algebraic stability of singly diagonally implicit Runge-Kutta (SDIRK) methods. The existing SDIRK methods are shown to be inapplicable to construct LN-stable ROW methods for numerical integration of stiff systems of ordinary differential equations.     

On high order strong stability preserving runge-kutta and multi step time discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.     

Fourth-order runge-kutta schemes for fluid mechanics applications

Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier-Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectrum. Implicit-Explicit (IMEX) schemes are used on separable problems that naturally partition into stiff and nonstiff components. Non-separable problems are solved with fully implicit schemes, oftentimes the implicit portion of an IMEX scheme. The convection-diffusion-reaction (CDR) equations allow a term by term stiff/nonstiff partition that is often well suited for IMEX methods. Major variables in CDR converge at near design-order rates with all formulations, including the fourth-order IMEX additive Runge-Kutta (ARK₂) schemes that are susceptible to order reduction. The semi-implicit backward differentiation formulae and IMEX ARK₂ schemes are of comparable efficiency. Laminar and turbulent aerodynamic applications require fully implicit schemes, as they are not profitably partitioned. All schemes achieve design-order convergence rates on the laminar problem. The fourth-order explicit singly diagonally implicit Runge-Kutta (ESDIRK4) scheme is more efficient than the popular second-order backward differentiation formulae (BDF2) method. The BDF2 and fourth-order modified extended backward differentiation formulae (MEBDF4) schemes are of comparable efficiency on the turbulent problem. High precision requirements slightly favor the MEBDF4 scheme (greater than three significant digits). Significant order reduction plagues the ESDIRK4 scheme in the turbulent case. The magnitude of the order reduction varies with Reynolds number. Poor performance of the high-order methods can partially be attributed to poor solver performance. Huge time steps allowed by high-order formulations challenge the capabilities of algebraic solver technology.     

Construction and Implementation of General Linear Methods for Ordinary Differential Equations: A Review

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge–Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge–Kutta stability is also briefly discussed     

GPG-stability of Runge-Kutta methods for generalized delay differential systems

The GP_G-stability of Runge-Kutta methods for the numerical solutions of the systems of delay differential equations is considered. The stability behaviour of implicit Runge-Kutta methods (IRK) is analyzed for the solution of the system of linear test equations with multiple delay terms. After an establishment of a sufficient condition for asymptotic stability of the solutions of the system, a criterion of numerical stability of IRK with the Lagrange interpolation process is given for any stepsize of the method.     

Preconditioning in parallel Runge-Kutta methods for stiff initial value problems

From a theoretical point of view, Runge-Kutta methods of collocation type belong to the most attractive step-by-step methods for integrating stiff problems. These methods combine excellent stability features with the property of superconvergence at the step points. Like the initial-value problem itself, they only need the given initial value without requiring additional starting values, and therefore, are a natural discretization of the initial-value problem. On the other hand, from a practical point of view, these methods have the drawback of requiring in each step the solution of a system of equations of dimension sd, s and d being the number of stages and the dimension of the initial-value problem, respectively. In contrast, linear multistep methods, the main competitor of Runge-Kutta methods, require the solution of systems of dimension d. However, parallel computers have changed the scene and have motivated us to design parallel iteration methods for solving the implicit systems in such a way that the resulting methods become efficient step-by-step methods for integrating stiff initial-value problems.