The field of numerical methods for complex physical systems is witnessing significant advancements, driven by the need for more accurate and efficient simulations. Recent developments are focused on improving the stability, accuracy, and robustness of numerical schemes, particularly in the context of multiphysics problems, such as fluid-structure interaction, magnetoelasticity, and elasto-acoustic wave propagation.

Notable progress is being made in the development of high-order methods, such as the hybrid high-order (HHO) methods, which offer improved accuracy and flexibility for simulating complex systems. Additionally, advancements in stabilization techniques, like local projection stabilization (LPS) methods, are enabling more robust and reliable simulations.

Innovative approaches, such as the maximum likelihood discretization (MLD) and the fast-wave slow-wave spectral deferred correction methods (FWSW-SDC), are being explored to address challenges in simulating complex phenomena, like transport equations and compressible Euler equations.

Some noteworthy papers in this area include:

• The paper on maximum likelihood discretization, which introduces a novel approach to discretizing transport equations, providing a more accurate and robust framework for simulating complex systems.
• The work on the coupled HDG discretization for the interaction between acoustic and elastic waves, which presents a new method for simulating multiphysics problems with improved accuracy and stability.