Convergence of Spectral-Grid Method for Burgers Equation with Initial-Boundary Conditions

of the Rate of ConvergenceConvergence of Spectral-Grid Method for Burgers Equation with Initial-Boundary Conditions methods available in the scientific literature cannot reflect the physical behavior of the equations when the viscosity v goes to zero.The main goal of the article was to develop a numerical scheme to overcome the shortcomings of existing schemes.It has been established that the use of the homogeneous Haar wavelet is accurate, simple, fast, flexible, convenient and requires little computational effort.Novel and efficient approaches to tackling the onedimensional quasilinear Burgers equation were introduced in various studies.In 2, a method employing the nonlinear Cole-Hopf transformation to reduce the equation to a onedimensional diffusion equation was presented.This approach involved semi-discretizing the linearized diffusion equation via the method of lines, leading to a system of ordinary differential equations in time.Solving these equations involved using the method of inverse differentiation of different orders, with an accompanying analysis of numerical errors, primarily performed at moderate kinematic viscosity values.Additionally, a gridless technique for solving the Burgers equation in the "intrinsic" Hilbert space of the reproducing kernel was investigated in 3.This method involved constructing derivatives' discretizations based on interpolants of Newton's basis functions, featuring a variable scale for localized sets of nodes.Its primary advantage lay in generating numerous small matrices in overlapping areas of influence rather than a large collocation matrix, resulting in a sparse matrix.The method demonstrated efficiency, accuracy, and stability, particularly in flows with high Reynolds numbers.Further advancements included the presentation of fully implicit numerical schemes for solving both onedimensional and two-dimensional nonstationary Burgers equations in 4.Here, the equations were spatially discretized using a second-order finite difference method, transforming them into nonlinear systems of ordinary differential equations.Employing second-order inverse differentiation formulas enabled time advancement.Comparative analysis against exact solutions and other schemes showcased the simplicity, efficiency, and accuracy of the proposed schemes, even in scenarios involving large Reynolds numbers.Another study in 5 described into a weak L-stable scheme for integrating the Burgers equation.This involved utilizing explicit inverse Taylor polynomial approximations and interpolating Hermite polynomial approximations to derive a circuit and formulate a vectorbased formula to solve the Burger equation.Discussions revolved around the stability and convergence of this scheme.Moreover, in 6, new finite-difference schemes for the shallow water model, formulated as a viscous Burgers-Poisson system with periodic boundary conditions, were described.These schemes, belonging to the family of threelevel linearized finite difference methods, proved effective in numerical simulations for both viscous and inviscid problems.