Mani Mehra

An adaptive wavelet collocation method for the solution of partial differential equations on the sphere

A dynamic adaptive numerical method for solving partial differential equations on the sphere is developed. The method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids, and is an extension of the adaptive wavelet collocation method to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. An O(N) hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate Laplace-Beltrami, Jacobian and flux-divergence operators. The accuracy and efficiency of the method is demonstrated using linear and nonlinear examples relevant to geophysical flows. Although the present paper considers only the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by considering appropriate coarse approximations to the desired manifold (here we used the icosahedral approximation to the sphere at the coarsest level).

CUBIC SPLINE ADAPTIVE WAVELET SCHEME TO SOLVE SINGULARLY PERTURBED REACTION DIFFUSION PROBLEMS

In this paper, the collocation method proposed by Cai and Wang1 has been reviewed in detail to solve singularly perturbed reaction diffusion equation of elliptic and parabolic types. The method is based on an interpolating wavelet transform using cubic spline on dyadic points. Adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples are presented for elliptic and parabolic problems. The purposed method comes up as a powerful tool for studying singular perturbation problems in term of effective grid generation and CPU time.

A three-step wavelet Galerkin method for parabolic and hyperbolic partial differential equations

A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed. The scheme is third-order accurate in time and O(2−jp ) accurate in space. Unlike Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving non-linear problems. The compactly supported orthogonal wavelet bases D6 developed by Daubechies are used in the Galerkin scheme. The proposed scheme is tested with both parabolic and hyperbolic partial differential equations. The numerical results indicate the versatility and effectiveness of the proposed scheme.

Time-accurate solutions of Korteweg-de Vries equation using wavelet Galerkin method

In this study we propose a space and time-accurate numerical method for Korteweg-de Vries equation. In deriving the computational scheme, Taylor generalized Euler time discretization is performed prior to wavelet based Galerkin spatial approximation. This leads to the implicit system which can also be solved by explicit algorithms. Korteweg-de Vries equation is also solved by a operator splitting method using wavelet-Taylor-Galerkin approach. Asymptotic stability of the schemes are verified.

Wavelets and Differential Equations‐A short review

The applications of wavelet theory in numerical methods for solving differential equations are roughly 20 years old. In the early nineties people were very optimistic because it seemed that many nice properties of wavelets would automatically leads to efficient numerical method for differential equations. The reason for this optimism was the fact that many nonlinear partial differential equations (PDEs) have solution containing local phenomena (e.g. formation of shock, hurricanes) and interactions between several scales (e.g. turbulence particularly atmospheric turbulence because there is motion on a continuous range of length scales). Such solutions can be well represented in wavelet bases because of its nice properties few of them like compact support (locality in space) and vanishing moment (locality in scale). Furthermore, this early optimism has been already honored by many authors [1–6] working in this area since then. Nevertheless, there often remains a large gap between a theoretical wavelet paper and the needs of an applied mathematician. This paper is an attempt to bridge this gap by providing a short review on wavelet based numerical methods for differential equations. Most common numerical methods used for numerical solution of physical problems (mostly leads to partial differential equation) fall in to following classes.

TIME ACCURATE FAST THREE-STEP WAVELET-GALERKIN METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

We introduce the concept of three-step wavelet-Galerkin method based on the Taylor series expansion in time. Unlike the Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving nonlinear problems. Numerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of functions which are smooth, except for localized regions, up to any given accuracy are presented. Here numerical experiments deal with advection equation with the spiky solution in one dimension, two dimensions and nonlinear equation with a shock in solution in two dimensions. Numerical results indicate the versatility and effectiveness of the proposed scheme.

An Adaptive Multilevel Wavelet Solver for Elliptic Equations on an Optimal Spherical Geodesic Grid

An adaptive multilevel wavelet solver for elliptic equations on an optimal spherical geodesic grid is developed. The method is based on second-generation spherical wavelets on almost uniform optimal spherical geodesic grids. It is an extension of the adaptive multilevel wavelet solver [O. V. Vasilyev and N. K.-R. Kevlahan, J. Comput. Phys., 206 (2005), pp. 412-431] to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. A hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate the Laplace-Beltrami operator. The optimal spherical geodesic grid [Internat. J. Comput. Geom. Appl., 16 (2006), pp. 75-93] is convergent in terms of local mean curvature and has lower truncation error than conventional spherical geodesic grids. The overall computational complexity of the solver is

Wavelet based preconditioners for sparse linear systems

O(\mathcal{N})

KRYLOV SUBSPACE SOLVERS IN PARALLEL NUMERICAL COMPUTATIONS OF PARTIAL DIFFERENTIAL EQUATIONS MODELING HEAT TRANSFER APPLICATIONS

, where

Fast wavelet-Taylor Galerkin method for linear and non-linear wave problems

\mathcal{N}