| We first present the implementation of a well-conditioned hierarchical bases (the Aiffa basis) for one-dimensional, triangular and tetrahedral elements in the FEMLAB finite element software. Using the domain mesh information provided by FEMLAB, we found an unconventional way to maintain the continuity of solutions across the inter-element boundaries. The conditionings of the global stiffness matrices of several standard problems are compared with Lagrange bases and are smaller for all cases.; High-order methods are desirable and more efficient to resolve complicated fine structures in flow problems but with the side effect of generating spurious oscillations near solution discontinuities. We describe a strategy for detecting discontinuities and for limiting spurious oscillations near such discontinuities when solving hyperbolic systems of conservation laws by high-order discontinuous Galerkin methods. The approach is based on the strong superconvergence property of the solutions at the outflow boundary of each element in smooth regions of the flow. By detecting discontinuities in such variables as density or entropy, limiting may be applied only in these regions; thereby, preserving a high order of accuracy in regions where solutions are smooth. Several one- and two-dimensional flow problems illustrate the performance of these approaches.; The last part addresses efficiency of time marching algorithms for transient problems. It is shown that an implicit DG scheme is more efficient than an explicit scheme. This efficiency is more pronounced when applied to steady state solutions. We apply the DG methods for the system of ordinary differential equations after spatial discretization for time marching. Besides its inherent A-stable property, another distinct advantage of this approach is the arbitrary high former order of accuracy O&parl0;&parl0;Dt&parr0;2p+1&parr0; when a pth-degree of polynomial basis is applied for time integration. The approximate system of linear equations arising from each Newton iteration is solved by the GMRES (generalized minimum residual) algorithm with a LU-SGS (lower-upper symmetric Gauss-Seidel) preconditioner. The Jacobian matrix is approximated by the difference quotient, rendering the method "matrix-free". The numerical results obtained indicate the significant improvement for time efficiency. (Abstract shortened by UMI.)... |
