Space-Time Methods for Hyperbolic Conservation Laws

Two major challenges for computational fluid dynamics are problems that involve wave propagation over long times and problems with a wide range of amplitude scales. An example with both of these characteristics is the propagation and generation of acoustic waves, where the mean-flow amplitude scales are typically orders-of-magnitude larger than those of the generated acoustics. Other examples include vortex evolution and the direct simulation of turbulence. All of these problems require greater than second-order accuracy, whereas for nonlinear equations, most current methods are at best second-order accurate. Of the higher-order (greater than second-order) methods that do exist, most are tailored to high-spatial resolution, coupled with time integrators that are only second or third-order accurate. But for wave phenomena, time accuracy is as important as spatial accuracy.