The recent developments in the research area of partial differential equations (PDEs) and their numerical solutions have shown a strong emphasis on structure-preserving methods and innovative parallelisation techniques. There is a notable trend towards high-order, mass, and energy-conserving schemes, particularly for nonlinear Schrödinger equations and related systems. These methods aim to maintain the physical invariants of the systems they model, ensuring accurate and reliable numerical solutions. Additionally, the use of representation theory for parallelising PDE solutions has opened new avenues for efficient computation, leveraging symmetry-adapted bases to decouple linear systems. Machine learning is also being integrated into the development of numerical methods, offering potential for more efficient and convergent solutions within limited computational budgets. Notably, explicit, energy-conserving particle-in-cell schemes and Riemannian optimisation methods for Bose-Einstein condensates have been introduced, showcasing advancements in both theoretical understanding and practical application.

Noteworthy Papers:

• A novel high-order, mass and energy-conserving scheme for the regularized logarithmic Schrödinger equation demonstrates precise conservation properties and computational efficiency.
• The variational derivation and compatible discretizations of the Maxwell-GLM system introduce new structure-preserving numerical methods with potential applications in general relativity.
• The first fully explicit symmetric low-regularity integrators for the nonlinear Schrödinger equation significantly reduce computational cost while preserving structure.
• Riemannian optimisation methods for multicomponent Bose-Einstein condensates offer efficient computation of ground states with robust convergence properties.