Waves are all around us, ranging from surface water waves, to sound waves, to gravitational waves. Mathematically, waves are described by hyperbolic PDEs. One of the most significant challenges in modeling waves arises from the fact that the solutions to nonlinear hyperbolic PDEs spontaneously develop ‘shocks’, or discontinuities. This has important implications both for the behavior of waves and for the computational methods used to model that behavior. I will discuss numerical tools for wave propagation that allow us to compute highly accurate solutions without producing spurious errors in the presence of shocks. I will also show how these tools have been applied to predict and understand the generation of elastic solitary waves in non-dispersive periodic materials.
