##
**The numerical solution of differential-algebraic systems by Runge-Kutta methods.**
*(English)*
Zbl 0683.65050

Lecture Notes in Mathematics, 1409. Berlin etc.: Springer-Verlag. vii, 139 p. DM 25.00 (1989).

In this monograph the authors systematically study application of implicit and semi-explicit Runge-Kutta methods for discrete approximation to differential-algebraic systems of ordinary differential equations. Applications to singularly perturbed systems are also included. The systems are classified according to the permutation index of the system, i.e. the behavior of the residual under a small perturbation of the solution.

It is shown that the appropriate choice of the Runge-Kutta method depends directly on the index of the system. Explicit choices are discussed for indices \(m=1,2,3\) (systems in normal form have index \(m=0)\). Numerical results are presented for a number of interesting applications. A new code RADAU 5 in Fortran is presented for a 3-stage Radau IIA Runge-Kutta method which is applicable for indices \(m=1,2,3.\)

Given a system of differential equations in implicit form \(F(Y',Y)=0\) the equation has perturbation index m along a solution Y on [0,b] if for all Z near Y the defect \(\delta (x)=F(Z',Z)\) satisfies \[ \| Z(x)- Y(x)\| \leq C(\| Z(0)-Y(0)\| +\sum^{m-1}_{j=0}\max_{0\leq t\leq x}\| \delta^{(j)}(t)\|). \] The authors employ the following canonical forms for systems with indices \(m=1,2,3:\) 1) \(m=1\), \(y'=f(y,z),\quad g(y,z)=0\) and \(g_ z\) is invertible. 2) \(m=2\), \(y'=f(y,z),\quad g(y)=0,\) and \(g_ yf_ z\) has a bounded inverse. 3) \(m=3\), \(y'=f(y,z),\quad z'=k(y,z,u),\quad 0=g(y,u)\) and \(g_ yf_ zk_ u\) has a bounded inverse. Which Runge-Kutta algorithm may be applied to a system depends on which of these forms it assumes. The applicable implicit Runge-Kutta methods are classified following J. C. Butcher [The numerical analysis of ordinary differential equations (1987; Zbl 0616.65072)] and K. Dekker and J. G. Verwer [Stability of Runge-Kutta methods for stiff nonlinear differential equations (1984; Zbl 0571.65057)] according to whether Gauss, Radau, or Lobatto quadrature is used. The specific algorithms considered are Gauss, Radau IA, IIA, and Lobatto IIIA, IIIC. In addition singly diagonal methods of R. Alexander [SIAM J. Numer. Anal. 14, 1006-1021 (1977; Zbl 0374.65038)] and of S. P. NĂ¸rsett and P. Thomsen [BIT 26, 100-113 (1986; Zbl 0627.65082)] are covered.

The notion of perturbation index and the classification of differential- algebraic systems is presented in chapter 1. Chapter 2 contains a review of Runge-Kutta methods for ordinary differential and differential- algebraic equations including methods for singularly perturbed systems. Chapters 3-6 select and describe methods appropriate respectively for indices \(m=1,2,3\). Proofs of convergence of the methods are presented including convergence for associated perturbed systems. Chapter 7 presents a simplified Newton method applicable to the relevant algorithms and chapter 8 concerns local error estimation. In chapter 9 a summary of numerical results is given for a number of interesting applications and chapter 10 presents the code RADAU 5.

The material presented in the monograph represents recent research by many authors which is published here for the first time in a unified form. A comprehensive bibliography and index are included.

It is shown that the appropriate choice of the Runge-Kutta method depends directly on the index of the system. Explicit choices are discussed for indices \(m=1,2,3\) (systems in normal form have index \(m=0)\). Numerical results are presented for a number of interesting applications. A new code RADAU 5 in Fortran is presented for a 3-stage Radau IIA Runge-Kutta method which is applicable for indices \(m=1,2,3.\)

Given a system of differential equations in implicit form \(F(Y',Y)=0\) the equation has perturbation index m along a solution Y on [0,b] if for all Z near Y the defect \(\delta (x)=F(Z',Z)\) satisfies \[ \| Z(x)- Y(x)\| \leq C(\| Z(0)-Y(0)\| +\sum^{m-1}_{j=0}\max_{0\leq t\leq x}\| \delta^{(j)}(t)\|). \] The authors employ the following canonical forms for systems with indices \(m=1,2,3:\) 1) \(m=1\), \(y'=f(y,z),\quad g(y,z)=0\) and \(g_ z\) is invertible. 2) \(m=2\), \(y'=f(y,z),\quad g(y)=0,\) and \(g_ yf_ z\) has a bounded inverse. 3) \(m=3\), \(y'=f(y,z),\quad z'=k(y,z,u),\quad 0=g(y,u)\) and \(g_ yf_ zk_ u\) has a bounded inverse. Which Runge-Kutta algorithm may be applied to a system depends on which of these forms it assumes. The applicable implicit Runge-Kutta methods are classified following J. C. Butcher [The numerical analysis of ordinary differential equations (1987; Zbl 0616.65072)] and K. Dekker and J. G. Verwer [Stability of Runge-Kutta methods for stiff nonlinear differential equations (1984; Zbl 0571.65057)] according to whether Gauss, Radau, or Lobatto quadrature is used. The specific algorithms considered are Gauss, Radau IA, IIA, and Lobatto IIIA, IIIC. In addition singly diagonal methods of R. Alexander [SIAM J. Numer. Anal. 14, 1006-1021 (1977; Zbl 0374.65038)] and of S. P. NĂ¸rsett and P. Thomsen [BIT 26, 100-113 (1986; Zbl 0627.65082)] are covered.

The notion of perturbation index and the classification of differential- algebraic systems is presented in chapter 1. Chapter 2 contains a review of Runge-Kutta methods for ordinary differential and differential- algebraic equations including methods for singularly perturbed systems. Chapters 3-6 select and describe methods appropriate respectively for indices \(m=1,2,3\). Proofs of convergence of the methods are presented including convergence for associated perturbed systems. Chapter 7 presents a simplified Newton method applicable to the relevant algorithms and chapter 8 concerns local error estimation. In chapter 9 a summary of numerical results is given for a number of interesting applications and chapter 10 presents the code RADAU 5.

The material presented in the monograph represents recent research by many authors which is published here for the first time in a unified form. A comprehensive bibliography and index are included.

Reviewer: J.B.Butler, jun

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65H10 | Numerical computation of solutions to systems of equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

34A34 | Nonlinear ordinary differential equations and systems |

34E15 | Singular perturbations for ordinary differential equations |