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VSMRK: A Parallel Implementation and the Performance of Variable Stepsize Multistep Runge-Kutta Methods for Stiff ODEs
H. Suhartanto,K. Burrage
Asian Journal of Information Technology , 2012,
Abstract: Many natural phenomena and applications in industry can be modeled as systems of Stiff Initial Value Problems (IVPs) of Ordinary Differential Equations (ODEs). Usually the problems to be solved are huge in dimension, hence require huge computing resources and time. This study describes the parallel implementation of Variable stepsize Multistep Runge-Kutta (MRK) method of Radau Type for solving stiff IVPs of ODEs and its performance on SGI Origin 2000. The numerical results show the superiority of the code compared to various standard code on dense and large sparse problems.
Stepsize Selection in Explicit Runge-Kutta Methods for Moderately Stiff Problems  [PDF]
Justin Steven Calder Prentice
Applied Mathematics (AM) , 2011, DOI: 10.4236/am.2011.26094
Abstract: 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.
One-Stage Implicit Rational Runge-Kutta Schemes for Treatment of Discontinous Initial Value Problems
P.O. Babatola,R.A. Ademiluyi,E.A. Areo
Journal of Engineering and Applied Sciences , 2012,
Abstract: This study describes one-stage Implicit Rational Runge-Kutta scheme for treatment of discontinuous ordinary differential equations. Its development adopts power series expansion method (Taylor and Binomial). The analysis of its basic properties uses Dalhquist model test equation. The results show that the schemes are consistent, convergent and A-stable. Numerical computations and comparative analysis with some standard methods show that the new schemes are efficient and accurate.
Exponential Runge-Kutta methods for stiff kinetic equations  [PDF]
Giacomo Dimarco,Lorenzo Pareschi
Mathematics , 2010,
Abstract: We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The methods are based on a decomposition of the collision operator into an equilibrium and a non equilibrium part and are exact for relaxation operators of BGK type. For Boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques.

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.
Implicit-Explicit Runge-Kutta schemes for hyperbolic systems with stiff relaxation and applications  [PDF]
Sebastiano Boscarino,Giovanni Russo
Mathematics , 2013,
Abstract: In this paper we give an overview of Implicit-Explicit Runge-Kutta schemes applied to hyperbolic systems with stiff relaxation. In particular, we focus on some recent results on the uniform accuracy for hyperbolic systems with stiff relaxation [6], and hyperbolic system with diffusive relaxation [7, 5, 4]. In the latter case, we present an original application to a model problem arising in Extended Thermodynamics.
Symmetric Uniformly Accurate Gauss-Runge-Kutta Method  [cached]
Dauda G. YAKUBU,Samaila MARKUS,Amina HAMZA,Abubakar M. KWAMI
Leonardo Journal of Sciences , 2007,
Abstract: Symmetric methods are particularly attractive for solving stiff ordinary differential equations. In this paper by the selection of Gauss-points for both interpolation and collocation, we derive high order symmetric single-step Gauss-Runge-Kutta collocation method for accurate solution of ordinary differential equations. The resulting symmetric method with continuous coefficients is evaluated for the proposed block method for accurate solution of ordinary differential equations. More interestingly, the block method is self-starting with adequate absolute stability interval that is capable of producing simultaneously dense approximation to the solution of ordinary differential equations at a block of points. The use of this method leads to a maximal gain in efficiency as well as in minimal function evaluation per step.
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.
A Comparative Study on Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Runge Kutta Methods  [PDF]
Md. Amirul Islam
American Journal of Computational Mathematics (AJCM) , 2015, DOI: 10.4236/ajcm.2015.53034
Abstract: This paper mainly presents Euler method and fourth-order Runge Kutta Method (RK4) for solving initial value problems (IVP) for ordinary differential equations (ODE). The two proposed methods are quite efficient and practically well suited for solving these problems. In order to verify the ac-curacy, we compare numerical solutions with the exact solutions. The numerical solutions are in good agreement with the exact solutions. Numerical comparisons between Euler method and Runge Kutta method have been presented. Also we compare the performance and the computational effort of such methods. In order to achieve higher accuracy in the solution, the step size needs to be very small. Finally we investigate and compute the errors of the two proposed methods for different step sizes to examine superiority. Several numerical examples are given to demonstrate the reliability and efficiency.
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.
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