## Preprint 2018-19

## Antonio Baeza, Raimund Bürger, María Carmen Martí, Pep Mulet, David Zorío:

### Approximate implicit Taylor methods for ODEs

### Abstract:

In this work an approximate version of the implicit Taylor methods for the solution of initial-value problems for systems of ordinary differential equations (ODEs) is introduced. The approach is based on the idea of a previous work on explicit Taylor methods [A. Baeza, S. Boscarino, P. Mulet, G. Russo and D. Zorıo, Approximate Taylor methods for ODEs, Computers and Fluids, 159 (2017), 156–166] and produces a method which only requires evaluations of the function that defines the ODE and its first order derivative, in contrast with the exact version that requires as many derivatives as the order of the method indicates. The numerical solution of the implicit equation that arises from the discretization by means of the Newton method is analyzed. The resulting algorithm is simpler to implement and has better performance than its exact counterpart.