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

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Kareem Taha Elgindy
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This paper presents an extension to the Cole-Hopf barycentric Gegenbauer integral pseudospectral (PS) method (CHBGPM) presented in [Elgindy, Kareem T., and Sayed A. Dahy. "High‐order numerical solution of viscous Burgers' equation using a Cole‐Hopf barycentric Gegenbauer integral pseudospectral method." Mathematical Methods in the Applied Sciences 41.16 (2018): 6226-6251.] to solve an initial-boundary value problem of Burgers' type when the boundary function k defined at the right boundary of the spatial domain vanishes at a finite set of real numbers or on a single/multiple subdomain(s) of the solution domain. We present a new strategy that is computationally more efficient than that presented in the above reference in the former case, and can be implemented successfully in the latter case when the method of the above reference fails to work. Moreover, fully exponential convergence rates are still preserved in both spatial and temporal directions if the boundary function k is sufficiently smooth. Numerical comparisons with other traditional methods in the literature are presented to confirm the efficiency of the proposed method. A numerical study of the condition number of the linear systems produced by the method is included.
Sayed A Dahy
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We present a novel, high-order numerical method to solve viscous Burger's 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 (GG) 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 paramter $\nu \to 0$, in the absence of any adaptive strategies typically required for adaptive refinements.