Current state-of-the-art flow simulations rely on four major techniques (finite difference, finite volume, finite element, or discontinuous galerkin methods) to solve flow equations. The methodologies are performing very well for medium size flow problems, but their performance drops appreciably for very large size flow problems (>1billion cells). This is partly caused by the limitation of the hardware/software to handle monumental amount of data generated by these billion computational cells. However, most of the flow solutions may not require all of these billion cells to provide sufficient accuracy. The fundamental flaw is the lack of efficient grid adaptation capability that can maintain specified error limit while minimizing computational resources. Oftentimes, the adapted grid is optimal for certain variables at specific scales but sub-optimal for others, which leads to unnecessary increase in numbers of computational variables after the adaptation cycle. To remedy this problem, a new paradigm for flow solver is developed with capabilities to optimally compress flow data and identify/resolve flow features at various scales of interest while maintaining accuracy. This is achieved through the development of a novel multi-scale flow solver that applies wavelet decomposition of the flow variables to provide optimal error control and grid adaptation capabilities as well as flow data compression. The wavelet solver is developed using finite difference and finite volume approaches to solve linear and non-linear 1D partial differential equations. Varieties of algorithms, boundary conditions, initial conditions are used to demonstrate its versatility and flexibility. The results showed that the wavelet solver was capable of providing significant compression (averaging between 70-90%) in flow data, while maintaining solution accuracy and adaptively resolving multi-scale flow features. The numerical error could be controlled by adjusting an error threshold limit, which dictated the amount of data compression and multi-scale adaptivity necessary to maintain the error. It was also observed that the wavelet compression did not alter the error characteristic of the underlining scheme.
NomenclatureA = Low resolution wavelet signal (Approximation) A i = Low resolution wavelet signal (Approximation) at i th wavelet level D = High resolution wavelet signal (Detail) D i = High resolution wavelet signal (Detail) at i th wavelet level N = number of equations/unknowns c = convection speed u , û = flow variable and/or signal in physical domain E u ,ˆE u = even indexed flow variable and/or signal in physical domain O u ,ˆO u = odd indexed flow variable and/or signal in physical domain k i u = flow variable at i th spatial location and k th time step k U = vector of flow variables at all spatial locations and k th time step k W = vector of wavelet variables at all frequency scales and k th time step x = spatial coordinate/distance from oscillating plate ε = exclusion threshold limit (wavelet variables) φ = forward wavelet transforma...
