A two step method for the numerical integration of stiff differential equations

2000 ◽  
Vol 73 (3) ◽  
pp. 333-340 ◽  
Author(s):  
H Bulut ◽  
M Inc
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Stiff Differential Equations ◽  
Step Method

The development of a shock from standing waves of finite amplitude in an isentropic fluid

1964 ◽  
Vol 60 (1) ◽  
pp. 129-135
Author(s):  
N. O. Weiss
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Integration ◽  
Standing Waves ◽  
Finite Amplitude ◽  
Step Method ◽  
Partial Differential ◽  
Simple Step

AbstractNumerical integration of the partial differential equations shows that rigid boundaries promote the development of a shock from a finite-amplitude disturbance. A simple step-by-step method agrees with one that also utilizes the characteristics.

Divergence of solutions of polynomial finite-difference equations

2012 ◽  
Vol 142 (4) ◽  
pp. 787-804 ◽  
Author(s):  
Harry Gingold
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Difference Equations ◽  
Step Method ◽  
Quadratic Equations ◽  
Finite Difference Equations ◽  
Order Polynomial ◽  
Independent Parameters

A theorem is proven for kth-order polynomial finite-difference equations that guarantees the divergence of solutions. A ‘basin of divergence’ is characterized and an order of divergence is provided. The basin of divergence is shown to depend on k independent parameters. An unconventional compactification method is used. Applications include the multi-step method in the numerical integration of ordinary differential equations, quadratic equations and the Henon map.

Efficient numerical integration of stiff differential equations in polymerisation reaction engineering: Computational aspects and applications

2012 ◽  
Vol 90 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Iván Zapata-González ◽  
Enrique Saldívar-Guerra ◽  
Antonio Flores-Tlacuahuac ◽  
Eduardo Vivaldo-Lima ◽  
José Ortiz-Cisneros
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Reaction Engineering ◽  
Stiff Differential Equations ◽  
Computational Aspects

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.

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