Solving Burgers’ equations with a new Haar wavelet approach: An application of Kronecker product and vectorization

In this work, Haar wavelet methods are developed for the numerical solution of the two-dimensional Burgers’ equation and a system of Burgers’ equations. The highest derivatives involved in the governing equation are approximated using Haar wavelets, while temporal discretization is achieved using several schemes: the θ-weighted scheme, the fourth-order Runge–Kutta (RK4) scheme, the adaptive step-size Runge–Kutta schemes including Runge–Kutta 3(2) (RK32) and the Runge–Kutta–Fehlberg (RKF45) as well as a strong stability-preserving Runge–Kutta (SSPRK(3,3)) scheme. To the best of our knowledge, this is the first matrix Haar wavelets formulation of the RK4, the RK32, the RKF45 and the SSPRK(3,3) schemes to two-dimensional time-dependent partial differential equations (PDEs). To reduce the size and complexity of the method, certain wavelet coefficients are deliberately omitted to reduce the size of the algebraic system of equations. The Kronecker product and vectorization operators are employed in the Runge–Kutta schemes to significantly reduce the computational cost of matrix operations. This strategy enhances the efficiency of the Runge–Kutta schemes compared to both the θ-weighted scheme and other methods reported in the literature. The proposed methods are assessed in terms of accuracy, spatial convergence rate, and computational cost. Comparative results demonstrate that the proposed schemes are both robust and computationally efficient.