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Contractivity of Runge-Kutta Methods for Multidelay Differential Equations

ZHANG Cheng-Jian,LIAO Xiao-Xin,

数学物理学报(A辑) , 2001,
Abstract: This paper deal with the contractivity of theoretical and numerical solutions for systems of multidelay differential equations(MDDEs), for which some new stability concepts, such as BN -stability and GRNm., -stability, are introduced. The approach yields that the BN -stability of the Runge-Kutta (RK) method and the corresponding continuous interpolant guarantee the contractivity (i. e. GRNm,. -stability) of the induced methods for MDDEs.
Multirate generalized additive Runge Kutta methods  [PDF]
Michael Guenther,Adrian Sandu
Computer Science , 2013,
Abstract: This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge-Kutta (GARK) methods (Sandu and Guenther, 2013). Multirate schemes use different step sizes for different components and for different partitions of the right-hand side based on the local activity levels. We show that the new multirate GARK family includes many well-known multirate schemes as special cases. The order conditions theory follows directly from the GARK accuracy theory. Nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold.
A class of generalized additive Runge-Kutta methods  [PDF]
Adrian Sandu,Michael Guenther
Computer Science , 2013,
Abstract: This work generalizes the additively partitioned Runge-Kutta methods by allowing for different stage values as arguments of different components of the right hand side. An order conditions theory is developed for the new family of generalized additive methods, and stability and monotonicity investigations are carried out. The paper discusses the construction and properties of implicit-explicit and implicit-implicit,methods in the new framework. The new family, named GARK, introduces additional flexibility when compared to traditional partitioned Runge-Kutta methods, and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales, or driven by multiple physical processes.

Zhang Chengjian,Jin Jie,

计算数学 , 2007,
Abstract: This paper is concerned with nonlinear stability and computational effectiveness of Runge-Kutta methods for solving stiff multi-delay integro-differential equations.The classi- cal Runge-Kutta methods,together with the compound quadrature formulae and the Pouzet quadrature formulae,are adapted to a class of nonlinear stiff multi-delay integro-differential equations of Volterra type.The analysis derive that the extended Runge-Kutta methods are globally and asymptotically stable under the suitable conditions.Moreover,the numerical experiments show that the presented methods are highly effective.
Some Stability and Convergence of Additive Runge-Kutta Methods for Delay Differential Equations with Many Delays
Haiyan Yuan,Jingjun Zhao,Yang Xu
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/456814
Abstract: This paper is devoted to the stability and convergence analysis of the additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of a delay differential equation with many delays. GDN stability and D-Convergence are introduced and proved. It is shown that strongly algebraically stability gives D-Convergence DA, DAS, and ASI stability give GDN stability. Some examples are given in the end of this paper which confirms our results.
Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise  [PDF]
Kristian Debrabant
Mathematics , 2009, DOI: 10.1007/s10543-010-0276-2
Abstract: A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order Runge-Kutta scheme for weak approximation, the new class of methods affords less random variable evaluations and is also applicable to SDEs with multidimensional noise. Order conditions up to order three are calculated and coefficients of a four stage third order method are given. This method has deterministic order four and minimized error constants, and needs in addition less function evaluations than the method of Platen. Applied to some examples, the new method is compared numerically with Platen's method and some well known second order methods and yields very promising results.
Construction of Additive Semi-Implicit Runge-Kutta methods with low-storage requirements  [PDF]
Inmaculada Higueras,Teo Roldán
Mathematics , 2015, DOI: 10.1007/s10915-015-0116-2
Abstract: Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be used to integrate the stiff terms while efficient explicit methods can be used for the non-stiff part of the problem. However, for systems with a large number of equations, memory storage requirement is also an important issue. When the high dimension of the problem compromises the computer memory capacity, it is important to incorporate low memory usage to some other properties of the scheme. In this paper we consider Additive Semi-Implicit Runge-Kutta (ASIRK) methods, a class of implicitexplicit Runge-Kutta methods for additive differential systems. We construct two second order 3-stage ASIRK schemes with low-storage requirements. Having in mind problems with stiffness parameters, besides accuracy and stability properties, we also impose stiff accuracy conditions. The numerical experiments done show the advantages of the new methods.
基于Additive Runge-Kutta方法的激波聚焦起爆高精度数值模拟
Numerical Application of Additive Runge-Kutta Methods on Detonation Initiation with Convergence of Shock Waves

- , 2016, DOI: 10.15918/j.tbit1001-0645.2016.02.006
Abstract: 基于详细氢氧化学动力学模型,建立了描述氢氧爆轰的多组分反应欧拉方程组. 针对建立的反应欧拉方程组,数值方法上采用3阶Additive Runge-Kutta方法对时间项进行积分,采用5阶精度的加权本质振荡(WENO)格式对空间对流项进行离散,自主研发了大规模高精度计算程序. 该程序能够处理化学反应源项引起的刚性问题,且能节省计算时间和计算内存. 对半球型、半椭球型、圆锥型3种结构形式凹面腔内的激波聚焦起爆过程进行了数值模拟,数值模拟研究得到了不同结构形式凹面腔内的激波聚焦起爆过程.
Based on detailed hydrogen and oxygen chemical kinetics model, the multi-species reactive Euler equations were established to describe the hydrogen and oxygen detonation. Using third-order Additive Runge-Kutta methods in time discretization, using fifth-order weighted essentially non-oscillatory (WENO) scheme in spatial discretization, a high-solution large-scale program was developed. It was verified that the program could solve stiff source term caused by multispecies chemical reaction, which could also save computation time and computation memory. With this program, the process of detonation initiation with convergence of shock waves was numerically simulated for 3 types of concave cavity, hemispherical concave cavity, semi-ellipsoidal concave cavity and conical concave cavity. The simulation results show the characteristic of detonation initiation in different types of concave cavity.
A Class of New Pouzet-Runge-Kutta-Type Methods for Nonlinear Functional Integro-Differential Equations
Chengjian Zhang
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/642318
Abstract: This paper presents a class of new numerical methods for nonlinear functional-integrodifferential equations, which are derived by an adaptation of Pouzet-Runge-Kutta methods originally introduced for standard Volterra integrodifferential equations. Based on the nonclassical Lipschitz condition, analytical and numerical stability is studied and some novel stability criteria are obtained. Numerical experiments further illustrate the theoretical results and the effectiveness of the methods. In the end, a comparison between the presented methods and the existed related methods is given.
Bigeometric Calculus and Runge Kutta Method  [PDF]
Mustafa Riza,Bu??E Emina?A
Mathematics , 2014,
Abstract: The properties of the Bigeometric or proportional derivative are presented and discussed explicitly. Based on this derivative, the Bigeometric Taylor theorem is worked out. As an application of this calculus, the Bigeometric Runge-Kutta method is derived and is applied to academic examples, with known closed form solutions, and a sample problem from mathematical modelling in biology. The comparison of the results of the Bigeometric Runge-Kutta method with the ordinary Runge-Kutta method shows that the Bigeometric Runge-Kutta method is at least for a particular set of initial value problems superior with respect to accuracy and computation time to the ordinary Runge-Kutta method.
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