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B-Theory of runge-kutta methods for stiff Volterra functional differential equations

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Science China Mathematics 2003年第5期46卷 662-674页

作者：李寿佛 Department of Mathematics Xiangtan University Xiangtan 411105 China

B-stability and B-convergence theories of runge-kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof runge-kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differentialequations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs ofother type which appear in practice.

关键词： stiff functional differential equations, runge-kutta methods, B-stability, B-con-vergence.

Algebraic dynamics algorithm:Numerical comparison with runge-kutta algorithm and symplectic geometric algorithm

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Science China(Physics,Mechanics & Astronomy) 2007年第1期50卷 53-69页

作者： WANG ShunJin ZHANG Hua Center of Theoretical Physics Sichuan UniversityChengdu 610064China

Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with runge-kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the runge-kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.

关键词： algebraic dynamics algorithm for ordinary differential equations preserving both geometrical and dynamical fidelity numerical comparison with runge-kutta algorithm and symplectic geometric algorithm

Symplectic partitioned runge-kutta method based onthe eighth-order nearly analytic discrete operator and its wavefield simulations

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Applied Geophysics 2014年第1期11卷 89-106,117,118页

作者：张朝元马啸杨磊宋国杰大理学院数学与计算机学院大理671003 清华大学数学科学系北京100084 西南石油大学理学院成都610500 清华大学计算机系北京100084

We propose a symplectic partitioned runge-kutta （SPRK） method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete （NAD） operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal *** stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction （LWC） method and the eighth-order staggered-grid （SG）method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical *** numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic explora

关键词： Symplectic partitioned runge-kutta method Nearly analytic discrete operator Numerical dispersion Wavefield simulation

基于runge-kutta的NURBS曲线实时前瞻插补算法

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华南理工大学学报（自然科学版） 2017年第10期45卷 121-128页

作者：吴玉香王鹏华南理工大学自动化科学与工程学院广东广州510640

针对传统的NURBS曲线加工过程中插补算法插补参数计算精度低、实时性不高以及加速度过大对机床造成的冲击大的问题,提出了基于runge-kutta的NURBS曲线实时前瞻插补算法.该算法采用经典runge-kutta方法计算插补参数,基于弓高误差和法向... 详细信息

针对传统的NURBS曲线加工过程中插补算法插补参数计算精度低、实时性不高以及加速度过大对机床造成的冲击大的问题,提出了基于runge-kutta的NURBS曲线实时前瞻插补算法.该算法采用经典runge-kutta方法计算插补参数,基于弓高误差和法向加速度约束条件自动调整进给速度,根据进给步长预期值与实际值的偏差进行参数校正.由粗插补得到的离线数据寻找进给速度极值点,并对曲线进行前瞻分段,找到各前瞻插补区间上的首末速度敏感点.根据敏感速度与插补距离之间的关系重新进行加减速控制,避免速度急剧变化,从而满足机床的加减速性能要求.最后,通过Matlab仿真验证了算法的有效性.

关键词：数控系统 NURBS曲线 runge-kutta 前瞻插补算法

runge-kutta Schemes Coefficients Simulation for Comparison and Visual Effects

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Engineering（科研） 2013年第5期5卷 530-536页

作者： Tajudeen Abiola Ogunniyi Salau Olusegun Olufemi Ajide Department of Mechanical Engineering University of Ibadan Ibadan Nigeria

runge-kutta scheme is one of the versatile numerical tools for the simulation of engineering systems. Despite its wide and acceptable engineering use, there is dearth of relevant literature bordering on visual impression possibility among different schemes coefficients which is the strong motivation for the present investigation of the third and fourth order schemes. The present study capitalise on results of tedious computation involving Taylor series expansion equivalent supplemented with Butcher assumptions and constraint equations of well-known works which captures the essential relationship between the coefficients. The simulation proceeds from random but valid specification of two out of the total coefficients possible per scheme. However the remaining coefficients are evaluated with application of appropriate function relationship. Eight and thirteen unknown coefficients were simulated respectively for third and fourth schemes over a total of five thousand cases each for relevant distribution statistics and scatter plots analysis for the purpose of scheme comparison and visual import. The respective three and four coefficients of the slope estimate for the third and fourth schemes have mix sign for large number of simulated cases. However, none of the two schemes have above three of these coefficients lesser than zero. The percentages of simulation results with two coefficients lesser than zero dominate and are respectively 56.88 and 77.10 for third and fourth schemes. It was observed that both popular third and fourth schemes belong to none of the coefficients being zero classification with respective percentage of 0.72 and 3.28 intotal simulated cases. The comparisons of corresponding scatter plots are visually exciting. The overall difference between corresponding scatter plots and distribution results can be used to justify the accuracy of fourth scheme over its counterpart third scheme.

关键词： runge-kutta Visual Effects Simulation Coefficients Taylor Series and Butcher Assumptions

runge-kutta法求解弹性地基梁的数值方法及其在基坑施工分析中的应用

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岩石力学与工程学报 2003年第Z1期22卷 2328-2332页

作者：谭新丁万涛山东大学岩土与结构工程研究中心济南250061

采用特殊处理的有限差分法求解弹性地基梁的挠曲方程,剪应力连续的区域用 runge-kutta 差分法离散,不连续的奇异点根据内力平衡条件建立传递矩阵.由于直接将要求的内力作为未知量,因而能处理复杂的边界条件,求解结果具有较高的精度.用... 详细信息

采用特殊处理的有限差分法求解弹性地基梁的挠曲方程,剪应力连续的区域用 runge-kutta 差分法离散,不连续的奇异点根据内力平衡条件建立传递矩阵.由于直接将要求的内力作为未知量,因而能处理复杂的边界条件,求解结果具有较高的精度.用该方法对基坑施工的工程进行了数值模拟.

关键词：土力学弹性地基梁 runge-kutta 传递矩阵基坑

Stepsize Selection in Explicit runge-kutta Methods for Moderately Stiff Problems

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Applied Mathematics 2011年第6期2卷 711-717页

作者： Justin Steven Calder Prentice 不详

We present an algorithm for determining the stepsize in an explicit runge-kutta method that is suitable when solving moderately stiff differential equations. The algorithm has a geometric character, and is based on a pair of semicircles that enclose the boundary of the stability region in the left half of the complex plane. The algorithm includes an error control device. We describe a vectorized form of the algorithm, and present a corresponding MATLAB code. Numerical examples for runge-kutta methods of third and fourth order demonstrate the properties and capabilities of the algorithm.

关键词： Moderately Stiff Problems runge-kutta Stepsize Jacobian Stability Region

The　Stability　of　runge－kutta　Methods　for　Systems　of　Delay　Differential　Equations

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Journal of Harbin Institute of Technology(New Series) 1996年第1期3卷 1-6页

作者：王晓彪刘明珠

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关键词： ss:Delay differential equations numerical solution runge-kutta methods Stability

Delay-dependent stability analysis of runge-kutta methods for neutral delay differential equations

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Journal of Harbin Institute of Technology(New Series) 2002年第2期9卷 129-135页

作者：宋明辉刘明珠 B S SIDIBE Dept.of Mathematics Harbin Institute of TechnologyHarbin 150001China

The aim of this paper is to study the asymptotic stability properties of runge kutta(R-K) methods for neutral differential equations(NDDEs) when they are applied to the linear test equation of the form: y′(t)=ay(t)+by(t-τ)+cy’(t-τ), t>0, y(t)=g(t), -τ≤t≤0, with a,b,c∈[FK(W+3mm\.3mm][TPP129A,+3mm?3mm,BP], τ>0 and g(t) is a continuous real value function. In this paper we are concerned with the dependence of stability region on a fixed but arbitrary delay τ. In fact, it is one of the *** open problems to investigate the delay dependent stability analysis for NDDEs. The results that the 2,3 stages non natural R-K methods are unstable as Radau IA and Lobatto IIIC are proved. And the s stages Radau IIA methods are unstable, however all Gauss methods are compatible.

关键词： neutral delay differention equation natural runge-kutta methods Nт(0)-stability Nт(0)-com patibility