This dissertation explores innovative strategies for solving linear variable-coefficient wave propagation problems in one and
multiple dimensions. The research uncovers a substantial contrast between two primary methodologies used to achieve high
formal accuracy: one approach enlarges the computational stencil, while the other enriches it with additional information,
such as derivatives or moments. The findings reveal that enrichment techniques are notably more effective in enhancing the
accuracy of numerical solutions. As a result, a Hermitian version of the Active Flux method is developed, which significantly
advances the state-of-the-art in solving wave propagation problems