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 → 0, in the absence of any adaptive strategies typically required for adaptive refinements.

Optimal Periodic Control of Unmanned Aerial Vehicles Based on Fourier Integral Pseudospectral and Edge-Detection Methods

“…The following notations are used throughout this paper to abridge and simplify the mathematical formulas. Many of these notations appeared earlier in [39,44]; however, for convenience and to keep the paper self-explanatory, we summarize them below together with the new notations. Logical Symbols.…”

“…We refer to Θ 2π by the "basic/principle" FIM, because of its simplest form, the "generating" FIM due to Formula (4), or the "natural" FIM, since its merits place it as the natural choice among all possible FIMs. For further information about FPS interpolation and quadratures, the reader may consult the recent works [39,42,44].…”

“…Moreover, the extreme thrust values are given, but the thrust optimal switching times are not known a priori. For a comprehensive survey on the excellent virtues of integral reformulations and FIMs used in our work, we refer to [39][40][41][42][43][44] and the references therein. Clear expositions of the IPS methods can also be found in [45][46][47][48][49][50] and the references therein.…”

A direct integral pseudospectral method for solving a class of infinite-horizon optimal control problems using Gegenbauer polynomials and certain parametric maps

“…The current methods do not suffer from this limitation, as they generally remain stable for large mesh grids and certain parameter ranges, as we will demonstrate later in Sections 5 and 9. For a comprehensive review on IPS and IS methods, the reader may consult [7][8][9][10][11][12][13], and the references therein.…”

A high-order embedded domain method combining a Predictor–Corrector-Fourier-Continuation-Gram method with an integral Fourier pseudospectral collocation method for solving linear partial differential equations in complex domains

“…A comprehensive review on recent progress on Fourier PS methods can be found in [2,[16][17][18][19]. For a survey on Gegenbauer polynomials and quadratures and their relevant theory, the reader may consult [13,[20][21][22][23], and the references therein.…”