The recent developments in the research area of numerical methods for partial differential equations (PDEs) have shown a strong focus on enhancing stability, accuracy, and efficiency of computational schemes. A notable trend is the integration of implicit-explicit (IMEX) frameworks, which have been extended to handle variable density and viscosity in incompressible flows, offering linear solutions and unconditional stability. Another significant advancement is the introduction of fully-discrete Lyapunov-consistent discretizations for convection-diffusion-reaction systems, which ensure stability properties are preserved in numerical simulations. Structure-preserving discontinuous Galerkin schemes have also been innovated for the Cahn-Hilliard equation, incorporating time adaptivity to manage distinct time scale dynamics. Additionally, robust time-discontinuous Galerkin methods combined with finite or virtual elements have been developed for the advection-diffusion equation, demonstrating optimal convergence rates and stability across different regimes.

Noteworthy papers include one proposing an IMEX framework for incompressible flows with variable density and viscosity, offering linear solutions and unconditional stability. Another paper introduces a fully-discrete Lyapunov-consistent discretization framework for convection-diffusion-reaction systems, ensuring stability in numerical simulations.