Multiresolution Solution of Burgers Equation with B-spline Wavelet Basis

TL;DR

This paper introduces a multiresolution numerical method using B-Spline wavelets for solving the Burgers equation, demonstrating improved accuracy with fewer basis functions compared to traditional methods.

Contribution

The paper develops a novel mixed numerical approach combining wavelet collocation with finite difference for nonlinear PDEs, specifically applied to Burgers equation.

Findings

Higher accuracy with fewer basis functions.

Effective handling of boundary conditions.

Superior performance over traditional methods.

Abstract

This paper represents a mixed numerical method for the multi-resolution solution of non-linear partial differential equations based on B-Spline wavelets. The method is based on a second-order finite difference formula combined with the collocation method which uses the wavelet basis and applied to the Burgers equation. Performance and accuracy of the numerical solutions are studied using three standard test cases with Dirichlet and Neumann boundary conditions. Comparing the results with other methods such as the fully implicit finite difference method and mixed finite difference/boundary element method shows a greater accuracy while using the smaller number of basis functions.