Runge-Kutta Implicit Methods

In this section we introduce the concept of implicit Runge-Kutta methods of collocation type without aiming for completeness. The use of implicit Runge-Kutta methods is quite recent in multibody dynamics. There are higher order A-stable methods in this class for simulating highly oscillatory problems as well as methods specially designed for constrained multibody systems, see Sec. 5.4.2. 

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D. Janezic and B. Orel. Implicit Runge-Kutta method for molecular dynamics integration. J. Chem. Info. Comp. Sd., 33 252-257, 1993. 

Janezic, D., Orel, B. Implicit Runge-Kutta Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 33 (1993) 252-257 Janezic, D., Orel, B. Improvement of Methods for Molecular Dynamics Integration. Int. J. Quant. Chem. 51 (1994) 407-415... 

Ascher, U. M., and Petzold, L. R., Projected implicit Runge-Kutta methods for differential algebraic equations, SIAM J. Num. Anal. 28(4), 1097 (1991). 

Petzold, L. R., Order results for implicit Runge-Kutta methods applied to differential/algebraic," SIAM Journal on Numerical Analysis, No. 4, pp. 837-852 (1986). 

M72 Solution of stiff differential equations semi-implicit Runge-Kutta method with backsteps Rosenbrock-Gottwa1d-Wanner 7200 7416... 

The basic formula of the semi-implicit Runge-Kutta methods is similar to... 

IIIN) Prokopakis, G. J., Seider, W. D. Adaptive Semi-implicit Runge-Kutta Method for Solu-... 

Fourth-order two-stage implicit Runge-Kutta method... 

An extension of semi-implicit Runge-Kutta methods is the Rosenbrock-Wanner method. In this method eq. (10.63) is replaced by the expression... 

Gustafsson K. 1993. Stepsize selection in implicit Runge-Kutta methods viewed as a control problem. 

To illustrate the principle this device is based on, let us consider the following two-terms semi-implicit Runge-Kutta method ... 

Implicit and Diagonally Implicit Runge-Kutta Methods... 

In the BzzMath library, classes based on implicit and diagonally implicit Runge-Kutta methods are not implemented to handle ODE problems with initial conditions. 

Let us emphasize that, while a typical RK method is not symplectic, some implicit Runge-Kutta methods are symplectic. The precise condition that must be... 

Michelsen s third order semi-implicit Runge-Kutta method is a modified version of the method originally proposed by Caillaud and Padmanabhan (1971). This third-order semi-implicit method is an improvement over the original version of semi-implicit methods proposed in 1963 by Rosenbrock. 

Caillaud, J.B., and L. Padmanabhan, An Improved Semi-Implicit Runge-Kutta Method for Stiff Systems, Chem. Eng. J. 2, 22 -2i2 (1971). 

Weimer, A.W., and D.E. Clough, A Critical Evaluation of the Semi-Implicit Runge-Kutta Methods for Stiff Systems, AIChEJ. 25, 730-732 (1979). 

Equations 4.25a, 4.25b are known as Hessenberg index-2 form. Codes suitable for numerical solution of such DAE systems are, e.g. DASPK 3.1 [19, 20] (based on the BDF method) and Radau5 [21] that uses an implicit Runge-Kutta method of order 5. 

Rosenbrock methods require the solution of a linear system of equations at each step, simplifying the solution of the problem compared to the implicit Runge-Kutta method. It is defined by... 

Cameron IT. Solution of Differential-Algebraic Systems using Diagonally Implicit Runge-Kutta Methods. IMA J of Numerical Analysis 1983 3 273 289. 

Williams R, Burrage RK, Cameron IT, Kerr M. A four-stage index 2 diagonally implicit Runge-Kutta method. Applied Numerical Mathematics 2002 40 415-432. 

Caillaud JB, Padmanabhan L (1971) An improved semi-implicit Runge-Kutta method for stiff systems. Chem Eng J 2 227. 

For stiff differential equations, an explicit method cannot be used to obtain a stable solution. To solve stiff systems, an unrealistically short step size h is required. On the other hand, the use of an implicit method requires an iterative solution of a nonlinear algebraic equation system, that is, a solution of k, from Equation A2.4. By a Taylor series development of y + /=i il i truncation after the first term, a semi-implicit Runge-Kutta method is obtained. The term k, can be calculated from [1]... 

When using the semi-implicit Runge-Kutta method, the calculation of the Jacobian matrix can be critical. The Jacobian, J, influences directly the values of the parameters k, and, thereby, the whole solution of y. An analytical expression of the Jacobian is always preferable over a numerical approximation. If the differentiation of the function is cumbersome, an approximation can be obtained with forward differences... 

After calculating k, according to Equation A2.4, is easily obtained from Equation A2.3. The coefficients an and bi for some semi-implicit Runge-Kutta methods are summarized in Table A2.1 [1-3]. 

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