Development of a Higher-Order Solver for Aerodynamic Applications

Abstract

We present the results from the development of a higher-order discontinuous Galerkin nite element solver using p-multigrid with line Jacobi smoothing. The line smoothing algorithm is presented for unstructured meshes, and p-multigrid is outlined for the non- linear Euler equations of gas dynamics. Analysis of 2-D advection shows the improved performance of line implicit versus point implicit relaxation. Through a mesh renemen t study, we determine that the accuracy of the discretization is the optimal O(h p+1 ) for three dieren t smooth problems. The multigrid convergence rate is found to be indepen- dent of the polynomial order but does depend weakly on the grid size. Timing studies for each problem indicate that higher order is advantageous over grid renemen t when high accuracy is required. HILE CFD has achieved signican t maturity during the past decades, computational costs are extremely large for aerodynamic simulations of aerospace vehicles. In this applied aerodynamics con- text, the discretization of the Euler and/or Navier- Stokes equations is almost exclusively performed by nite volume algorithms. The pioneering work of Jameson began this evolution to the status quo. 1{3 During the 1980's, upwinding mechanisms were in- corporated into these nite volume algorithms leading to increased robustness for applications with strong shocks, and perhaps more importantly, to better res- olution of viscous layers due to decreased numerical dissipation in these regions. 4{8 The 1990's saw major advances in the application of nite volume methods to Navier-Stokes simulations (in particular the Reynolds- Averaged Navier-Stokes equations). Signican t gains were made in the use of unstructured meshes and solu- tion techniques for viscous problems. 9{12 While these algorithmic developments have resulted in an ability to simulate aerodynamic o ws for very complex prob- lems, the time required to achieve sucien t accuracy in a reliable manner places a severe constraint on the application of CFD to aerospace design. The accuracies of many of the nite-v olume meth- ods currently used in aerodynamics are at best p = 2, i.e. the error decreases as O(h p ) where h is a measure of the grid spacing. As a practical matter, however, the accuracy of these methods on more realistic prob- lems appears to be less than this, ranging between 1 p 2. The development of a practical higher- order solution method could result in a signican t decrease in the computational time required to achieve an acceptable error level. To better demonstrate the