High-order Adaptive Methods for Convection-dominated Nonlinear Problems Using Multilevel Solution Techniques


The spatial discretization of nonlinear hyperbolic PDE to high order of accuracy on unstructured meshes can be accomplished by local discretization methods, such as the Discontinuous Galerkin (DG) method, or the Spectral Difference (SD) Method. While a high order spatial discretization enables one to achieve better resolution with fewer degrees of freedom, the overall efficiency and robustness of a numerical scheme for large-scale applications depends on the solution methodology for the (nonlinear) system of equations arising from the discretization. Furthermore, efficiency and robustness can be defined in various ways, trading of such constraints as convergence properties, stability, CPU time, memory requirements, ease of implementation, and parallelizability.

For such nonlinear problems as high-Reynolds-number compressible fluid flow, efficiency and robustness of high order methods is still lagging behind best-practice (lower order accurate) industrial tools. To improve the status quo, one may identify convergence acceleration and adaptivity as two key technologies. Convergence acceleration, such as multigrid methods combined with efficient relaxation techniques, is paramount in increasing the efficiency of any solver, and rather mature for first and second order methods, most notably finite volume schemes. While established concepts usually allow an extension to higher order methods, the constraints often change dramatically (e.g. storage requirements). In practice, for most problems of practical interest, such as steady-state problems in transonic aerodynamics, no high-order solver currently achieves a faster turnaround compared to current state-of-the-art second order methods.

Compressible fluids flow often produce discontinuities, such as compression shocks, which make it necessary to design high-order schemes with shock capturing capabilities, which complicates the problem enormously. Adaptive mesh refinement enables the capture of very small scales of shock waves by lower order approximations without degrading the overall accuracy. Adaptation methods based on wavelets, induced by a multisresolution analysis, are very attractive. While these techniques have been successfully used with finite-volume solvers, the extension to higher order DG discretizations using so-called multiwavelets is yet to be fully developed. The potential, however is tremendous: the multiscale analysis framework, formulated on a tree of nested meshes, allows a unified treatment of mesh adaptation and multigrid solution methods within a solver architecture equipped with a compact, algorithmically optimal data structure. Limiter functions may also be defined based on the regularity properties of the solution esablished by the multiresolution analysis.