Numerical solutions of multi-dimensional partial differential equations using ad adaptive wavelet method

In this work, we describe an adaptive wavelet method for the solution of time-independent and time-dependent partial differential equations in d-dimensions. The method is based on d-dimensional interpolating wavelets constructed from tensor products of 1-D interpolating wavelets. The connection between interpolating wavelets and dyadic grid points, and the fact that wavelet amplitudes indicate the local regularity of solutions are used in the construction of a computational grid of irregular points. Operations, such as the wavelet transform, its inverse, and interpolation are performed efficiently. In the spatial discretization, the derivative approximation on the irregular grid is obtained by means of consistent finite differences. An extension of the adaptive method to problems defined on more complicated domains is achieved by a domain transformation technique. For time-independent problems, the method is tested on 2- and 3-D Poisson and Helmholtz problems with exact manufactured solutions in order to numerically study the connection between the order of the wavelet, the order of finite difference, the threshold values, and the accuracy of numerical solutions. The combustion of a $2$-D flame ball-vortex interaction is used as a test problem for the time-independent algorithm. Application of adaptive method to incompressible Navier-Stokes equations is accomplished through the use of the Chorin projection method for time discretization. We apply the algorithm to simulate the flow in the 2-D lid-driven cavity at moderate Reynolds numbers and in the 2-D differentially-heated cavity at high Rayleigh numbers, and in the 3-D differentially heated cavity for various values of Rayleigh numbers. It is found that numerical results, while requiring a relatively small number of degrees of freedom, are in good agreement with the most accurate results available in the literature.