Aiaa 9993309 Eecient Implementations of the Quadrature-free Discontinuous Galerkin Method Efficient Implementations of the Quadrature-free Discontinuous Galerkin Method

The e ciency of the quadrature-free form of the discontinuous Galerkin method in two dimensions, and brie y in three dimensions, is examined. Most of the work for constant-coe cient, linear problems involves the volume and edge integrations, and the transformation of information from the volume to the edges. These operations can be viewed as matrix-vector multiplications. Many of the matrices are sparse as a result of symmetry, and blocking and specialized multiplication routines are used to account for the sparsity. By optimizing these operations, a 35% reduction in total CPU time is achieved. For nonlinear problems, the calculation of the ux becomes dominant because of the cost associated with polynomial products and inversion. This component of the work can be reduced by up to 75% when the products are approximated by truncating terms. Because the cost is high for nonlinear problems on general elements, it is suggested that simpli ed physics and the most e cient element types be used over most of the domain.