Finite difference Wavelet–Galerkin method for the numerical solution of elastohydrodynamic lubrication problems

Abstract

The use of compactly supported wavelet functions has become increasingly popular in the development of numerical solutions for differential equations, especially for problems with local high gradient. Daubechies wavelets have been successfully used as base functions in several schemes like the Wavelet–Galerkin Method, due to their compact support, orthogonality, and multi-resolution properties. Another advantage of wavelet-based methods is the fact that the calculation of the inner products of wavelet basis functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of approximating the integral by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness, mass and geometry matrices. Solving connection coefficients in the Wavelet–Galerkin Method is complicated. In this work, Finite Difference Wavelet–Galerkin Method is developed for the numerical solution of elastohydrodynamic lubrication (EHL) problems. This approach enables the use of the finite difference and multiresolution analysis. Using this method numerical solution (pressure (P) and film thickness (H)) of the EHL problem is obtained and presented in comparison with the finite difference solution in tables and figures. Finite difference Wavelet–Galerkin method is easier to implement than the Wavelet–Galerkin Method and it gives fast and accurate solution.