Runge-Kutta (RK) methods often exhibit order reduction when applied to stiff problems. This talk addresses this challenge, focusing on stiff problems across various fields. I will begin by introducing Diagonally Implicit Runge-Kutta (DIRK) methods with high Weak Stage Order (WSO), which effectively mitigate order reduction for linear problems with time-independent operators. Following this, I will discuss explicit RK methods with high WSO, tailored for initial-boundary value problems with time-dependent boundary conditions in hyperbolic systems. Additionally, I will present theoretical order barriers that connect the WSO of an RK scheme to its order and the number of stages for fully implicit RK, DIRK, and ERK schemes, providing a framework for designing schemes with high WSO.
