## Section: New Results

### Splitting methods with complex coefficients for some classes of evolution equations

Participant : Philippe Chartier.

We are concerned in [13] with the numerical solution obtained by splitting methods of certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. In a previous paper, Castella et al. demonstrated the possibility to overcome this second-order barrier by considering splitting methods with complex-valued coefficients and built up methods of orders 3 to 14 . In this paper, we reconsider the technique employed therein and show that it is inherently bound to order 14 and largely sub-optimal with respect to error constants. As an alternative, we solve directly the algebraic equations arising from the order conditions and construct several methods of orders 4 , 6 , 8 and 16 that are the most accurate ones available at present time.