The field of numerical methods for partial differential equations is rapidly evolving, with a focus on developing innovative and efficient algorithms for solving complex problems. Recent developments have seen a shift towards the use of adaptive and asymptotic-preserving schemes, which are capable of capturing complex phenomena and behavior in various fields, including fluid dynamics, biology, and materials science. Notably, researchers have made significant progress in the development of new finite difference methods, such as the zigzag schemes, which offer improved stability and accuracy. Additionally, there have been advances in the development of optimized Schwarz methods for heterogeneous heat transfer problems and novel direct algorithms for reconstructing small conductivity inclusions in subdiffusion models. These developments have the potential to greatly impact various fields, including engineering, physics, and biology, by enabling more accurate and efficient simulations of complex systems. Noteworthy papers include: The paper on An Adaptive-rank Approach with Greedy Sampling for Multi-scale BGK Equations, which proposes a novel adaptive-rank method for simulating multi-scale BGK equations. The paper on A new class of finite difference methods: The zigzag schemes, which introduces a novel class of finite difference approximations that offer improved stability and accuracy. The paper on Direct Algorithms for Reconstructing Small Conductivity Inclusions in Subdiffusion, which develops novel direct algorithms for reconstructing small conductivity inclusions in subdiffusion models.