  Runge-Kutta (RK) methods may exhibit order reduction when applied to certain
stiff problems. While fully implicit RK schemes exist that avoid order
reduction via high-stage order, DIRK (diagonally implicit Runge-Kutta) schemes
are practically important due to their structural simplicity; however, these
cannot possess high stage order. The concept of weak stage order (WSO) can also
overcome order reduction, and it is compatible with the DIRK structure. DIRK
schemes of WSO up to 3 have been proposed in the past, however, based on a
simplified framework that cannot be extended beyond WSO 3. In this work a
general theory of WSO is employed to overcome the prior WSO barrier and to
construct practically useful high-order DIRK schemes with WSO 4 and above. The
resulting DIRK schemes are stiffly accurate, L-stable, have optimized error
coefficients, and are demonstrated to perform well on a portfolio of relevant
ODE and PDE test problems.
