Abstract:
Wavelets are compact functions in space and time which enable easy data com pression through multi-resolution analysis. The compression is done by scattering dif ferent resolution levels of wavelets onto the domain, and then discarding the wavelets with small energies. Using these properties, wavelets can be an efficient and easily applied tool to construct an adaptive grid for the solution of a Partial Differential Equation (PDE) with local structures. In this work, wavelets are used in this manner of compressing the interpolated data, being combined with finite difference discretiza tion to solve the PDE. To calculate the spatial derivatives of the compressed interpo lation, algebraic polynomial fits and cubic splines are used on the irregular adaptive grid. These two approaches are compared with each other and finite differences on regular grids. Various problems in 1-D and 2-D are solved. As model problems, Pois son’s Equation and Helmholtz Equation are solved with artificially created Gaussian Pulse as the solution. The results seemed to be in agreement in terms of the order of error with the known exact solutions of the model problems. A new application for wavelet optimized finite differences is also suggested. To that purpose, split-step (projection-correction) time scheme is implemented for the Navier-Stokes Equations governing infamous lid-driven cavity problem. The qualitative results seemed to be in agreement with other results in literature, however the method is observed to be not of any advantage for this problem as this problem does not have strong localized structures. PETSc framework provided the high-level tools, such as matrix and vector operations and linear solvers, and the work is conducted through this perspective.
