Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations

A discontinuous Galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions. For many practical problems, this approach requires fewer operations and less storage than conventional implementations but preserves the compactness and robustness that is inherent in the discontinuous Galerkin method. Test problems include the linear and nonlinear one-dimensional scalar advection of both smooth and discontinuous initial value problems, the two-dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity, and two-dimensional linear Euler solutions on unstructured grids.