A fast Newton-Krylov method has been developed to solve the Euler equations governing compressible inviscid flow on unstructured grids. Excellent convergence rates are achieved for subsonic and transonic flows over airfoils. Extension to turbulent flows is underway. We have also developed an agglomeration multigrid algorithm using directional coarsening. We have developed and applied such methods for the propagation and scattering of electromagnetic waves, i.e., solving the Maxwell equations in the time domain.
Numerical dissipation is added using a characteristic splitting.
The method has been implemented on curvilinear grids with a careful treatment of interfaces between different media. Results obtained are excellent, with substantial reductions in error relative to a fourth-order method. In the future, we plan to apply these methods to computational aeroacoustics. We are developing and investigating various algorithms for the design of aerodynamic configurations. The idea is to have the designer specify a multi-point objective function and various constraints, and the algorithm then finds the shape which minimizes the objective function while satisfying the constraints.