The field of numerical methods for complex systems is rapidly evolving, with a focus on developing efficient and accurate algorithms for solving large-scale problems. Recent developments have centered around improving the performance of existing methods, such as the variable-preconditioned transformed primal-dual method, and introducing new techniques, like the Skew Gradient Embedding framework. Notable advancements have also been made in the development of adaptive schemes for hyperbolic conservation laws and high-order numerical homogenization methods for multiscale systems. These innovations have the potential to significantly impact various fields, including fluid dynamics, materials science, and plasma physics.

Some noteworthy papers in this area include: The Variable-Preconditioned Transformed Primal-Dual method, which achieves superior computational efficiency compared to existing methods. The Skew Gradient Embedding framework, which provides a unified stabilization strategy for constructing numerical schemes that preserve energy dissipation rates or ensure discrete energy stability. The high-order numerical homogenization method, which enables efficient long-time simulations of multiscale systems.