Comparison of Consistent Integration Versus Adaptive Quadrature For Taming Aliasing Errors

Most spectral/hp element methods, whether employing a modal or nodal basis representation, evaluate non-linear difierential operators in "physical space" at a collection of collocation or quadrature points. The number of points used is often set to what is needed to represent the original solution over an element (or integrate the square of the function over an element), and not to what is needed to represent the square of the function. This discrepancy leads to aliasing errors, which when the flelds are highly resolved have little appreciative impact and hence can be ignored. In under-resolved scenarios, aliasing can pollute the solution leading to decreased accuracy and issues of stability. These errors can be eliminated by consistent integration at the price of increased computational cost. In most engineering simulations, however, the issue is not binary: not all elements within a simulation domain contain under-resolved solutions nor full-resolved solutions. The location and times at which elements support under-resolved solutions varies based upon the dynamics of the system. Hence an e-cient mean of taming aliasing errors can be through dynamic quadrature. In this report, we present analysis that compares the computational e-ciency of an adaptive consistent integration strategy that dynamically adapts the level of quadrature based upon aliasing indicator, with the traditional consistent integration approach. Hierarchical Gauss-Kronrod quadrature is used, allowing for both error estimation and consistent integration of quadratic non-linearities at a single set of points which have as their subset the classic integration points. Our theoretical estimates indicate for our adaptive scheme to be e-cient, linear quadrature needs to be used on the majority of elements to keep the overall computational cost less than the cost of the consistent integration approach. Two dimensional type elements are considered in all the analysis.