Efficent implementation of high order methods in computational fluid dynamics

Traditionally, finite element methods generate progressively higher order accurate solutions by use of higher degree trial space bases for the weak statement construc-tion. This invariably yields matrix equations of greater bandwidth thus increasing implementational and computational costs. A new approach to designing high order - defined here to exceed third - accurate methods has been developed and tested. The systematic construction of progressively higher order spatial approximations is achieved via a modified equation analysis, which allows one to clearly identify correction terms appropriate for a desired accuracy order. The resulting perturbed PDE is shown to be consistent with the Taylor Weak Statement formulation. It confirms the expected high order of spatial accuracy in TWS constructions and provides a highly efficient dispersion error control mechanism whose application is based on the specifics of the solution domain discretization and physics of the problem. A distinguishing desirable property of the developed method is solution matrix bandwidth containment, i.e. bandwidth always remains equal to that of the linear basis (second order) discretization. This permits combining the computational efficiency of the lower order methods with superior accuracy inherent in higher order approximations. Numerical simulations compare performance of the developed method to that of the GWS and TWS formulations. Uniform mesh refinement convergence results con-firm the order of truncation error for each method. High order formulation is shown to require significantly fewer nodes to accurately resolve solution gradients for con-vection dominated problems. Benchmark problem applications for the compressible Euler and incompressible Navier-Stokes equations complete the manuscript. In both cases the developed high order formulation is shown to result in more accurate solu-tions on coarser discretizations, thus preserving the design trends illustrated for the model advection-diffusion equation