Self-Adaptive Multilevel Methods for Fluid Flow Problems

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Journal Title, Volume, Page: 
Ph.D. Thesis Clarkson Univ., Potsdam, NY
Year of Publication: 
Anwar Saleh
Current Affiliation: 
Department of Mathematics, Faculty of Science, An-Najah National University, Nablus, Palestine
Preferred Abstract (Original): 

A computational study of self-adaptive multilevel methods for complex fluid flow problems is made to test the efficiency of these methods. The model problem is time-dependent, nonlinear, convective dominated, and diffusion-limited. Numerical solutions exist, although not multilevel or adaptive ones. A comparison of two adaptive multilevel methods, the multilevel adaptive technique (MLAT) and the fast adaptive composite grid method (FAC), is given to show the possible advantages of the FAC method over the MLAT method by applying them to solve a Poisson equation with an analytical solution. The model problem consists of the quasi-compressible system of the anelastic equations with an initial condition representing a negatively buoyant blob of cold air, which descends to the ground and spreads laterally forming a cold front. A multilevel solution is first obtained on a staggered grid using finite differencing both in time and space. Then the two self-adaptive multilevel methods (MLAT ant FAC) are applied. Numerical results are discussed and compared to numerical solution obtained by Fulton (18) using a Fourier-Chebyshev spectral method with a semi-implicit Runge-Kutta scheme for time integration. Numerical results, as expected, show the FAC is more accurate (at interface) than the MLAT when applied to a Poisson equation. When the self-adaptive versions of the FAC and the MLAT are applied to the complex anelastic equations, computational results show little difference between the two methods and a saving of up to 70 percent in execution time (compared to uniform grid methods of the same accuracy), and when compared to a Fourier-Chebyshev spectral model of the same problem, they are faster for modest accuracy, while the spectral method is faster for higher accuracy.