The recent developments in the research area of numerical methods for partial differential equations (PDEs) and kinetic equations show a strong focus on enhancing stability, accuracy, and computational efficiency. There is a notable trend towards the development of novel diagnostic tools and frameworks that can better handle discontinuities and high-dimensional problems. These advancements include the introduction of multiplicative dynamical low-rank approximations for kinetic equations, which significantly reduce computational effort, and the application of dual-pairing summation-by-parts finite difference frameworks for nonlinear conservation laws, ensuring entropy stability and high-order accuracy. Additionally, there is a growing interest in model order reduction techniques, such as slice sampling tensor completion, to address the complexities of parametric dynamical systems. The impact of numerical fluxes on high-order schemes is also being rigorously studied, with a focus on optimizing performance across different types of problems. Furthermore, the field is witnessing the development of asymptotic-preserving schemes for stiff systems, such as isentropic flow in pipe networks, and the theoretical analysis of reduced order models for fluid dynamics. These developments collectively push the boundaries of what is achievable in computational physics and engineering, offering new tools and insights for tackling complex PDEs and kinetic equations.

Noteworthy papers include one that introduces a novel diagnostic tool for evaluating the convection boundedness properties of numerical schemes across discontinuities, and another that presents a dual-pairing summation-by-parts finite difference framework for nonlinear conservation laws, ensuring entropy stability and high-order accuracy.