The field of numerical methods for solving hyperbolic conservation laws and singular perturbation problems is experiencing significant advancements. Researchers are developing innovative methods to improve the accuracy and efficiency of simulations, particularly in the context of moving grids and curvilinear space-time representations. One notable direction is the development of high-order methods that can achieve temporal superconvergence, reducing aliasing errors and improving the overall robustness of the simulations. Another area of focus is the optimal error analysis of finite element methods for complex problems, such as diffuse interface two-phase MHD flows. Noteworthy papers in this area include:

• High-order nodal space-time flux reconstruction methods that achieve temporal superconvergence and reduce aliasing errors.
• Optimal L2 error estimates of fully discrete finite element methods for 2D/3D diffuse interface two-phase MHD flows, which ensure mass conservation and unconditional energy stability.
• Robust PAMPA schemes in the DG formulation on unstructured triangular meshes, which provide bound preservation, oscillation elimination, and rigorous boundary conditions.