High‐order numerical solution of viscous Burgers' equation using a Cole‐Hopf barycentric Gegenbauer integral pseudospectral method

Abstract

We present a novel high‐order numerical method to solve viscous Burgers' equation with smooth initial and boundary data. The proposed method combines Cole‐Hopf transformation with well‐conditioned integral reformulations to reduce the problem into either a single easy‐to‐solve integral equation with no constraints or an integral equation provided by a single integral boundary condition. Fully exponential convergence rates are established in both spatial and temporal directions by embracing a full Gegenbauer collocation scheme based on Gegenbauer‐Gauss mesh grids using apt Gegenbauer parameter values and the latest technology of barycentric Gegenbauer differentiation and integration matrices. The global collocation matrices of the reduced algebraic linear systems were derived allowing for direct linear system solvers to be used. Rigorous error and convergence analyses are presented in addition to two easy‐to‐implement pseudocodes of the proposed computational algorithms. We further show three numerical tests to support the theoretical investigations and demonstrate the superior accuracy of the method even when the viscosity parameter ν→0, in the absence of any adaptive strategies typically required for adaptive refinements.