A new class of diagonally implicit Runge–Kutta methods with zero dissipation and minimized dispersion error

2020 ◽  
Vol 376 ◽  
pp. 112841 ◽  
Author(s):  
Subhajit Giri ◽  
Shuvam Sen
Keyword(s):  
Dispersion Error ◽  
New Class ◽  
Runge Kutta

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

scholarly journals Rosenbrock Type Methods for Solving Non-Linear Second-Order in Time Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2225
Author(s):  
Maria Jesus Moreta
Keyword(s):  
Computational Cost ◽  
Intermediate Stage ◽  
Second Order ◽  
First Order ◽  
New Class ◽  
Nyström Methods ◽  
Non Linear ◽  
Second Order In Time ◽  
Linear Problems ◽  
Runge Kutta

In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they can be seen as a generalization of these methods when they are applied to second-order in time problems which have been previously transformed into first-order in time problems. As they also follow the ideas of Runge–Kutta–Nyström methods when solving second-order in time problems, we have called them Rosenbrock–Nyström methods. When solving non-linear problems, Rosenbrock–Nyström methods present less computational cost than implicit Runge–Kutta–Nyström ones, as the non-linear systems which arise at every intermediate stage when Runge–Kutta–Nyström methods are used are replaced with sequences of linear ones.

scholarly journals Structure aware Runge–Kutta time stepping for spacetime tents

2020 ◽  
Vol 1 (4) ◽  
Author(s):  
Jay Gopalakrishnan ◽  
Joachim Schöberl ◽  
Christoph Wintersteiger
Keyword(s):  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Convergence Properties ◽  
New Class ◽  
Time Stepping ◽  
New Methods ◽  
Optimal Convergence Rates ◽  
Runge Kutta ◽  
Discrete Stability

Abstract We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.

scholarly journals Phase Fitted And Amplification Fitted Of Runge-Kutta-Fehlberg Method Of Order 4(5) For Solving Oscillatory Problems

2020 ◽  
Vol 17 (2(SI)) ◽  
pp. 0689
Author(s):  
Mohammed Salih ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Periodic Solution ◽  
Approximate Solution ◽  
Numerical Solution ◽  
Present Method ◽  
Phase Lag ◽  
Real Solution ◽  
Oscillatory Solutions ◽  
Dispersion Error ◽  
Recent Method ◽  
Runge Kutta

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg method.

scholarly journals A Class of Two-Derivative Two-Step Runge-Kutta Methods for Non-Stiff ODEs

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
I. B. Aiguobasimwin ◽  
R. I. Okuonghae
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Initial Value Problems ◽  
Initial Value ◽  
Large Interval ◽  
New Class ◽  
Second Derivatives ◽  
Stiff Odes ◽  
Runge Kutta ◽  
Special Case

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

scholarly journals EXTRAPOLATED IMPLICIT–EXPLICIT RUNGE–KUTTA METHODS

2014 ◽  
Vol 19 (1) ◽  
pp. 18-43 ◽  
Author(s):  
Angelamaria Cardone ◽  
Zdzislaw Jackiewicz ◽  
Adrian Sandu ◽  
Hong Zhang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
General Linear Methods ◽  
General Linear ◽  
Current Step ◽  
New Class ◽  
Runge Kutta ◽  
Previous Step

We investigate a new class of implicit–explicit singly diagonally implicit Runge–Kutta methods for ordinary differential equations with both non-stiff and stiff components. The approach is based on extrapolation of the stage values at the current step by stage values in the previous step. This approach was first proposed by the authors in context of implicit–explicit general linear methods.

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