A multi-resolution discontinuous galerkin method for unsteady compressible flows

• Main Author: Shelton, Andrew Brian
• Published: Georgia Institute of Technology 2008
• Subjects: High order; Shock; Wavelet; Adaptation; Vortex; Unsteady flow (Aerodynamics); Galerkin methods; Fluid dynamics
• Online Access: http://hdl.handle.net/1853/24715

The issue of local scale and smoothness presents a crucial and daunting challenge for numerical simulation methods in fluid dynamics. Yet in the interests of both accuracy and economy, how can one devise a general technique that efficiently resolves flow features of consequence and discriminates against others which are either ``negligible' or amenable to ``universal' modeling? This is particularly difficult because geometries of engineering interest are complex and multi-dimensional, precluding a priori knowledge of the flowfield. To address this challenge, the current work employs wavelet theory for the local scale decomposition of functions, which provides a natural mechanism for the adaptive compression of data. The resulting technique is known as the Multi-Resolution Discontinuous Galerkin (MRDG) method. This research successfully demonstrates that the multi-resolution framework and the discontinuous Galerkin method are well-suited for a new approach to accuracy and cost as demonstrated by the relative ease of their integration in spatial dimension greater than one. Some specific steps achieved include the implementation of suitable data encoding and compression algorithms, construction of multi-wavelet expansion bases in one and two dimensions, and derivation of the multi-resolution derivative operator that includes an upwind-type correction to the central scheme. Solutions with the MRDG method are observed to adapt to and track both smooth and discontinuous flow features in an entirely solution-driven manner without the need for a priori user knowledge of those flow features. Run-time efficiency and local adaptation characteristics are explored via a series of classic test problems.