The field of numerical methods for complex systems is moving towards the development of more accurate and efficient algorithms for solving problems with complex topology and geometry. Researchers are focusing on creating methods that can handle multiple materials, non-slip boundary conditions, and high-order convergence rates. The use of novel techniques such as micromorphic-based artificial diffusion and semi-Lagrangian finite volume methods is becoming increasingly popular. These methods are being applied to a range of problems, including convection-diffusion equations, acoustic scattering, and interface tracking.

Noteworthy papers include: The paper on fourth- and higher-order interface tracking of multiple materials, which proposes a multiphase cubic MARS method that achieves high-order accuracy and handles complex topology and geometry with ease. The paper on moment-enhanced shallow water equations, which proposes modified models that perform well under non-slip boundary conditions and are compatible with previous analysis on hyperbolicity. The paper on a micromorphic-based artificial diffusion method, which presents a novel approach to stabilizing finite element approximations of convection-diffusion problems. The paper on fourth- and higher-order semi-Lagrangian finite volume methods, which proposes a family of methods that feature high-order convergence rates and are applicable to complex domains. The paper on a time-frequency method for acoustic scattering with trapping, which introduces a Fourier transform method for solving the acoustic scattering problem in regimes with trapping regions.