Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

Holm, Bärbel; Wihler, Thomas (2018). Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up. Numerische Mathematik, 138(3), pp. 767-799. Springer 10.1007/s00211-017-0918-2

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We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Wihler, Thomas

Subjects:

500 Science > 510 Mathematics

ISSN:

0029-599X

Publisher:

Springer

Language:

English

Submitter:

Olivier Bernard Mila

Date Deposited:

17 Apr 2018 08:41

Last Modified:

24 Oct 2019 23:51

Publisher DOI:

10.1007/s00211-017-0918-2

BORIS DOI:

10.7892/boris.109184

URI:

https://boris.unibe.ch/id/eprint/109184

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