A multiresolution collocation method and its convergence for Burgers' type equations

In this article, a hybrid numerical method based on Haar wavelets and finite differences is proposed for shock ridden evolutionary nonlinear time-dependent partial differential equations (PDEs). A linear procedure using Taylor expansions is adopted to linearize the nonlinearity. The Euler difference scheme is used to discretize the time derivative part of the PDE. The PDE is converted into full algebraic form, once the space derivatives are replaced by finite Haar series. Convergence analysis is performed both in space and time, and computational stability of the proposed method is also discussed. Different benchmark cases related to 1-D and 2-D Burgers' type equations are considered to verify the theory. The proposed method is also compared numerically with existing methods in the literature.