The paper Functional-preserving predictor-corrector multiderivative schemes of Jochen Schütz, Eleni Theodosiou, and me has been published in Proceedings in Applied Mathematics and Mechanics.
In this work, we develop a class of high-order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite-Birkhoff-Predictor-Corrector methods and the technique of relaxation for numerical methods of ODEs. We explain the algorithm in detail and show numerical results for two- and three-derivative methods, comparing relaxed and unrelaxed methods. The numerical results demonstrate that, at the slight cost of the relaxation, an improved scheme is obtained.
As usual, you can find the preprint on arXiv.