The field of numerical methods for complex systems is witnessing significant advancements, with a focus on developing innovative and efficient schemes for solving various types of equations. Researchers are exploring new approaches to improve the stability, accuracy, and robustness of numerical methods, particularly for problems involving nonlinearities, singularities, and multiple scales. Notably, the development of unconditionally stable and conservative schemes is gaining attention, as these methods can handle challenging problems without requiring excessive computational resources. Furthermore, the application of optimization techniques and variational formulations is becoming increasingly popular, enabling the solution of complex problems in a more efficient and reliable manner. Some noteworthy papers in this regard include:

• A paper on a Gauge-Uzawa finite element method for the chemo-repulsion-Navier-Stokes system, which establishes a fully discrete projection framework with optimal error estimates.
• A work on a stochastic branching particle method for solving non-conservative reaction-diffusion equations, which yields a mesh-free and nonnegativity-preserving scheme.
• A study on a semi-Lagrangian conservative and unconditionally stable scheme for nonlinear advection-diffusion problems, which ensures conservation and stability without requiring excessive computational overhead.