Integration Of Second-order Differential Equations Research Articles

Efficient computation of the sinc matrix function for the integration of second-order differential equations

Efficient computation of the sinc matrix function for the integration of second-order differential equations

Embedded implicit Runge–Kutta Nyström method for solving second-order differential equations

An embedded diagonally implicit Runge–Kutta Nyström (RKN) method is constructed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. This embedded method is derived using a three-stage diagonally implicit RKN method of order four within which a third-order three stage diagonally implicit RKN method is embedded. We demonstrate how this system can be solved, and by an appropriate choice of free parameters, we obtain an optimized RKN(4,3) embedded algorithm. We also examine the intervals of stability and show that the method is strongly stable within an appropriate region of stability and is thus suitable for oscillatory problems by applying the method to the test equation y″=−ω2 y, ω>0. Necessary and sufficient conditions are given for this method to possess non-vanishing intervals of periodicity, for the fourth-order method. Finally, we present the coefficients of the method optimized for small truncation errors. This new scheme is likely to be efficient for the numerical integration of second-order differential equations with periodic solutions, using adaptive step size.

Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.'s

Panovsky and Richardson [A family of implicit Chebyshev methods for the numerical integration of second-order differential equations, J. Comput. Appl. Math. 23 (1988) 35–51] presented a method based on Chebyshev approximations for numerically solving the problem y ″ = f ( x , y ) , being the steplength constant. Coleman and Booth [Analysis of a Family of Chebyshev Methods for y ″ = f ( x , y ) , J. Comput. Appl. Math. 44 (1992) 95–114] made an analysis of the above method and suggested the convenience to design a variable steplength implementation. As far as we know this goal has not been achieved until now. Later on we extended the above method (this journal, 2003), and obtained a scheme for numerically solving the equation y ″ - 2 gy ′ + ( g 2 + w 2 ) = f ( x , y ) . The question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper.

Alternative characterization of time integration schemes

This work presents an alternative definition of the numerical angular frequency and numerical damping ratio, within the framework of direct time integration of second-order differential equations. The classical approach, based on the amplification matrix concept, deals with the ability of the numerical schemes to deliver an accurate response in time for problems with free vibration behaviour. It is well suited to the study of the schemes, both explicit and implicit, used in transient dynamic analysis. The alternative approach emphasizes the capability of the scheme to deliver an accurate resonant response for problems with forced vibration behaviour. Such frequency viewpoint is helpful in the study of the implicit schemes used in civil engineering applications.

Diagonally Implicit Runge–Kutta–Nyström Methods for Oscillatory Problems

Implicit Runge–Kutta–Nyström (RKN) methods are constructed for the integration of second-order differential equations possessing an oscillatory solution. Based on a linear homogeneous test model we analyse the phase errors (or dispersion) introduced by these methods and derive so-called dispersion relations. Diagonally implicit RKN methods of relatively low algebraic order are constructed, which have a high order of dispersion (up to 10). Application of these methods to a number of test examples (linear as well as nonlinear) yields a greatly reduced phase error when compared with “conventional” DIRKN methods.

A family of implicit Chebyshev methods for the numerical integration of second-order differential equations

A family of implicit methods based on intra-step Chebyshev interpolation is developed for the solution of initial-value problems whose differential equations are of the special second-order form y ″ = f ( y ( x); x). The general procedure allows stepsizes which are considerably larger than commonly used in conventional methods. Computation overhead is comparable to that required by high-order single or multistep procedures. In addition, the iterative nature of the method substantially reduces local errors while maintaining a low rate of global error growth.

De Vogelaere's method for the numerical integration of second-order differential equations without explicit first derivatives: Application to coupled equations arising from the schro¨dinger equation

De Vogelaere's method for the numerical integration of second-order differential equations without explicit first derivatives: Application to coupled equations arising from the schro¨dinger equation