To solve a time-dependent PDE numerically, one must consider an appropriate discretization both for space and time.  This can be especially challenging when considering nonlinear hyperbolic equations that may develop shocks.  This PhD project focuses on improving the time integration technique for hyperbolic equations.  It uses the inherent mathematical information from the spatial scheme, such as a discontinuous Galerkin finite element method to improve time integration.  This project will focus on mathematically reducing the errors, improving the cfl restriction and nonlinear equations.  Computationally efficient time-space pairings will be developed.  This is a U.S. Air Force Office of Scientific Research (AFOSR) project.