Extrapolation integrators for quasilinear implicit ODEs.

*(English)*Zbl 0617.65078
Large scale scientific computing, Proc. Meet., Oberwolfach/FRG 1985, Proc. Sci. Comput. 7, 37-50 (1987).

[For the entire collection see Zbl 0614.00022.]

This paper deals with quasilinear implicit ODEs of the form (1) \(B(y)y'=f(y)\). The main emphasis of the paper will be on problems, where B is nonsingular \((index=0)\). Extensions to problems with index \(=\) 1, where B is singular, are also included. In large scale scientific computing, problems of the type (1) may arise e.g. in chemical reaction kinetics, when thermodynamic equations are added, or in method of lines treatment for time-dependent PDEs with moving spatial grids.

At present, the numerical solution of (1) is typically attacked by BDF- type codes such as LSODI and DASSL. The latter code is also designed for the treatment of problems with index \(>0\). Recently, assocaited extrapolation techniques have been developed for: (a) problems (1) with index \(=\) 0 [cf. the first author, SIAM Rev. 27, 505-535 (1985; Zbl 0602.65047)], (b) problems (1) with constant B and index \(=\) 1 [cf. the first author, E. Hairer, J. Zugck: One-Step and Extrapolation Methods for Differential-Algebraic Systems. Univ. Heidelberg, SFB 123: Tech. Rep. 318]. It is the purpose of the present paper to report about recent progress made beyond the cited papers.

This paper deals with quasilinear implicit ODEs of the form (1) \(B(y)y'=f(y)\). The main emphasis of the paper will be on problems, where B is nonsingular \((index=0)\). Extensions to problems with index \(=\) 1, where B is singular, are also included. In large scale scientific computing, problems of the type (1) may arise e.g. in chemical reaction kinetics, when thermodynamic equations are added, or in method of lines treatment for time-dependent PDEs with moving spatial grids.

At present, the numerical solution of (1) is typically attacked by BDF- type codes such as LSODI and DASSL. The latter code is also designed for the treatment of problems with index \(>0\). Recently, assocaited extrapolation techniques have been developed for: (a) problems (1) with index \(=\) 0 [cf. the first author, SIAM Rev. 27, 505-535 (1985; Zbl 0602.65047)], (b) problems (1) with constant B and index \(=\) 1 [cf. the first author, E. Hairer, J. Zugck: One-Step and Extrapolation Methods for Differential-Algebraic Systems. Univ. Heidelberg, SFB 123: Tech. Rep. 318]. It is the purpose of the present paper to report about recent progress made beyond the cited papers.

##### MSC:

65L05 | Numerical methods for initial value problems |

34A34 | Nonlinear ordinary differential equations and systems, general theory |