PARALLEL WAVELET METHODS FOR COMPUTATIONAL FLUID DYNAMICS

NUMERICAL METHODS HAVE BEEN DEVELOPED FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, USING THE FAMILY OF COMPACTLY SUPPORTED WAVELET FUNCTIONS THAT ARE ORTHOGONAL. BEING BOTH COMPACTLY SUPPORTED AND ORTHOGONAL, THE WAVELETS COMBINE THE ADVANTAGES OF FINITE ELEMENT, SPLINE, AND FOURIER SPECTRAL METHODS. TO TEST THE FEASIBILITY OF THE WAVELET METHODS, PHENOMENA THAT EXHIBIT SINGULAR BEHAVIOR WERE EXAMINED. THESE INCLUDE SHOCKS (E.G. BURGERS EQUATION, THE GAS DYNAMICS/COMPRESSIBLE NAVIER-STOKES EQUATIONS) AND TURBULENCE (INCOMPRESIBLE NAVIER-STOKES EQUATIONS). THE VARIOUS TECHNICAL PROBLEMS THAT OCCUR IN THE PRACTICAL USE OF WAVELETS HAVE BEEN SOLVED, I.E., EXACT EVALUATION OF FUNCTIONALS AND BOUNDARY VALUE PROBLEMS. WAVELET BASED METHODS COULD PROVIDE A MAJOR ADVANCE IN THE FIELD OF COMPUTATIONAL FLUID DYNAMICS. IT IS EXPECTED THAT WAVELETS WILL BE PARTICULARLY USEFUL IN RESOLVING PHENOMENA RELATED TO TURBULENCE AND SHOCK DYNAMICS AND TO INTERFACE DISCONTINUITIES, I.E., ELECTROMAGNETIC AND ACOUSTIC, DIRECT, AND INVERSE PROBLEMS. PHASE I WILL DEVELOP TWO-DIMENSIONAL, PARALLEL VERSIONS OF THE WAVELET-GALERKIN ALGORITHM FOR THE EULER AND NAVIER-STOKES EQUATIONS. THESE WAVELET METHODS (WAVELET AND CAPACITANCE MATRIX METHODS) ARE DRIVEN BY THE PERIODIC SOLVERS FOR THE EULER AND NAVIER-STOKES EQUATIONS. ON SERIAL MACHINES CURRENT WAVELET METHODS CAN BE FASTER THAN THE FAST FOURIER TRANSFORM BASED SPECTRAL METHODS. THE COMPUTATIONAL BOTTLENECK IS IN THE EVALUATION OF THE NONLINEAR TERMS (90% OF CENTRAL PROCESSING UNIT TIME). HOWEVER, THE WAVELET EVALUATIONS, BECAUSE OF LOCAL SUPPORT OF THE WAVELET BASIS, ARE PERFECTLY PARALLEL. THEREFORE, IT WILL BE EXTREMELY EFFICIENT TO DEVELOP THESE METHODS ON, FOR INSTANCE, THE INTEL TOUCHSTONE PLATFORM.