The field of numerical methods for partial differential equations is witnessing significant developments, with a focus on designing innovative schemes that can efficiently handle complex problems. Researchers are exploring new approaches to ensure positivity preservation, high-order accuracy, and energy dissipation in various numerical methods, including direct discontinuous Galerkin schemes and exponential time differencing Runge-Kutta methods. Moreover, there is a growing interest in applying these methods to real-world problems, such as chemotaxis-Navier-Stokes systems and ideal magnetohydrodynamics. Noteworthy papers in this area include the development of a positivity-preserving hybrid DDG method for Poisson-Nernst-Planck systems, which ensures rigorous positivity preservation and high-order accuracy. Another significant contribution is the introduction of a comprehensive finite element approximation framework for three-dimensional Landau-de Gennes Q-tensor energies for nematic liquid crystals, which demonstrates optimal order convergence rate in the energy norm.