  In this work, we construct and derive a new class of exponentially fitted
two-derivative diagonally implicit Runge--Kutta (EFTDDIRK) methods for the
numerical solution of differential equations with oscillatory solutions. First,
a general format of so-called modified two-derivative diagonally implicit
Runge--Kutta methods (TDDIRK) is proposed. Their order conditions up to order
six are derived by introducing a set of bi-coloured rooted trees and deriving
new elementary weights. Next, we build exponential fitting conditions in order
for these modified TDDIRK methods to treat oscillatory solutions, leading to
EFTDDIRK methods. In particular, a family of 2-stage fourth-order, a
fifth-order, and a 3-stage sixth-order EFTDDIRK schemes are derived. These can
be considered as superconvergent methods. The stability and phase-lag analysis
of the new methods are also investigated, leading to optimized fourth-order
schemes, which turn out to be much more accurate and efficient than their
non-optimized versions. Finally, we carry out numerical experiments on some
oscillatory test problems. Our numerical results clearly demonstrate the
accuracy and efficiency of the newly derived methods when compared with
existing implicit Runge--Kutta methods and two-derivative Runge--Kutta methods
of the same order in the literature.
