The field of numerical methods for wave propagation and elasticity is moving towards the development of more efficient and accurate methods for modeling complex structures and phenomena. Researchers are exploring new approaches, such as quasi-Trefftz methods and grid-characteristic methods, to improve the accuracy and computational efficiency of numerical simulations. The use of these methods is allowing for the modeling of large and complex thin objects, such as buildings and space rockets, with a reasonable size calculation grid. Additionally, there is a focus on the mathematical and numerical analysis of the symmetry and positivity of tensor-valued spring constants, which is crucial for fracture simulations of elastic bodies. Noteworthy papers in this area include: The paper on quasi-Trefftz spaces for a first-order formulation of the Helmholtz equation, which is the first step in the development of quasi-Trefftz methods for first-order differential systems. The paper on the whys and hows of conditioning of DG plane wave Trefftz methods, which carefully examines the conditioning of the plane-wave discontinuous Galerkin method and presents results on preconditioning strategies.