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An efficient method for solving implicit and explicit stiff differential equations. (English) Zbl 0962.65057
This paper deals with an efficient algorithm which applies to both explicit as well as implicit ordinary differential equations. It differs from the traditional Runge-Kutta method. Using a polynomial of degree \(s\) for the solution \(y(x)\) leads to a nonlinear system which is solved by the Newton method. The formal precedure for uncoupling the algebraic system into a block-diagonal matrix with \(s\) blocks of size \(n\) is derived for any virtual number of stages \(s\).
The method may easily be constructed to be either \(A\)- or \(L\)-stable. In particular, for \(s= 3\) it has the same precision and stability properties as the well-known Runge-Kutta based Radau IIA method. Further, it may be considered as a good candidate for solving differential-algebraic equations of even higher index. Finally, the method is demonstrated by its application in the multibody dynamics.
MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34A09 Implicit ordinary differential equations, differential-algebraic equations
65L80 Numerical methods for differential-algebraic equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
Software:
PSIDE
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References:
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