Adaptive wavelet methods for PDEs and Integral equations; Prof. Dr. Mani Mehra (IIT Delhi)

ABSTRACT: Most of the times, while dealing with the physical problems of real world such as climate modelling and numerical weather prediction etc. one ends up with PDEs on the sphere. The subject of solving PDEs on the sphere using wavelets is not much developed. One of the work in this direction is a dynamic adaptive numerical method for solving PDEs on the sphere [Mehra and Kevlahan, 2008] using second generation spherical wavelet. The main difficulty with second generation wavelet is that an initial mesh structure is required to approximate the manifold. This difficulty can be handled with meshfree methods. In this work, we develop an adaptive meshfree diffusion wavelet method on the sphere (AMDWMS). Although the present work considers only the sphere, the strength of this method is that it can be easily extended to other manifolds. The beauty of the proposed method is that the diffusion operator is used both for the construction of the diffusion wavelet as well as for the approximation of the differential operator involved in the PDE at hand. Numerical results are given on couple of test problems. Moreover, the problem of pattern formation on the surface of the sphere (using Turing equations) is addressed to test the strength of the method. The numerical results show that the method can accurately capture the emergence of the localized patterns at all the scales and the node arrangement is accordingly adapted.