Wavelet and its variants based numerical methods for partial differential equations

Abstract

Since the 1990s, wavelet theory has been adopted for numerically solving the partial differential equations (PDEs). There is an immense research available on wavelet based techniques for numerically solving PDEs. But the theory of wavelet to numerically solve PDEs on arbitrary manifolds is yet in its emerging phase. Moreover, handling general boundary conditions using wavelets is also a tedious task. In this thesis, fast adaptive methods based on wavelets and their variants are developed which are easily extendable to general manifolds and can handle general boundaries. Second generation wavelet, spectral graph wavelet (SPGW) and curvelet are used for this purpose. Curvelet is already used in various areas of engineering, but to best of our knowledge it has very thin appearance in the field of PDEs. \par We started with Daubechies and second-generation wavelet to get a better understanding of wavelet-based methods for solving PDEs. Daubechies wavelet have been widely used for numerically solving PDEs, but we have used Daubechies wavelet for solving real life problems, $i.e.$, traffic flow problems. Furthermore, we developed a Matlab toolbox which contains the routines for the second generation wavelet transformation and inverse wavelet transformation on the space $\mathcal{L}_2([a,b])$. These wavelet transforms are further used for computing the wavelet and scaling function values ($\psi(x)$ and $\phi(x)$ respectively). We have also included the Matlab code for generating second generation wavelet based adaptive grid in our suite. \par As our aim was to tackle PDEs with different boundaries and Daubechies wavelet based methods are limited to periodic boundary conditions, we used second generation wavelet and third generation wavelet. The construction of second and third-generation wavelets is unaffected by boundaries because these wavelets are built directly on the space where one needs to tackle his/her problems. These wavelets can be used for solving all types of boundaries, however in case of classical wavelets like Daubechies wavelets, additional efforts are to be made to include all types of boundaries in addition to its basic construction. \par To be able to solve PDEs with solution having orientation features, we shifted to curvelets. Curvelets are attractive, because they effectively describe essential problems in which wavelet designs are far from ideals. The importance of curvelet is described in the following ways: \begin{itemize} \item Because of the bad orientation selectivity of wavelets, they do not present higher-dimensional irregularities efficiently. This makes curvelets interesting as they can yield an ideally adapted mathematical architecture to represent functions that display smooth punctuated curve. \item Tools from hereditary multi-scale analysis such as wavelets are insufficient to detect, establish or present a compact presentation of the intermediate dimensional arrangement. Curvelet is the better product because it produces better-adapted choices by consolidating concepts from geometry with ideas from classical multiscale analysis. \item Although there is wide research available for wavelets on arbitrary manifolds, yet the theory of wavelets for numerically solving PDEs on arbitrary manifolds is in its emerging phase. The most significant characteristic of the curvelet is that it can be designed on arbitrary manifolds. \end{itemize} \par Curvelet based fast adaptive technique is constructed for numerically solving PDEs on general manifolds. This method uses closest point form for approximating the Laplacian-Beltrami ($\triangledown^{2}$) operator and some suitable method (e.g., Crank Nicolson) for time integration of a given PDE. Curvelet is used for the following two purposes: \begin{itemize} \item For compression of the differential operators and hence for the rapid computation of the powers of the matrices associated with the PDE's numerical solution. \item Curvelet coefficients are used as an indicator function to guide the refinement of the node arrangement and in this way curvelet is used for adaptivity. \end{itemize} \par We have tested the curvelet based adaptive methods on many problems in one dimension, two dimension, three dimension and on the general manifolds. On the sphere, we have solved the non-linear Schr\"{o}dinger equation (NLS) which is a standout amongst the most important physics models with several applications in various fields, for example, self-focussing in laser pulses, non-linear optics, models of protein elements, plasma material science and so on. The experimental results in MATLAB shows that in terms of computational time, the developed technique is highly efficient. \par Altogether, the proposed methods are applied on a large number of test problems. Some of these test problems are elementary PDEs (Burger's equation) and some are more challenging problems (reaction-diffusion equation on the sphere, Schnakenberg model evolving on the surface of ellipsoid). These numerical tests reveal that the proposed methods can accurately capture the development of localized patterns on all scales and adapt the node arrangement accordingly. The CPU time analysis of the methods reveals that the methods are efficient as compared to the traditional methods. The convergence of the proposed methods is also numerically verified.