High‐order numerical solution of second‐order one‐dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method

Abstract

SUMMARYWe present a high-order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second-order onedimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well-conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out in order to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications. This is a preprint copy that has been accepted for publication in Numerical Methods for Partial Differential Equations.

Distributed optimal control of viscous Burgers' equation via a high‐order, linearization, integral, nodal discontinuous Gegenbauer‐Galerkin method

“…This is quite useful when conducting error analysis of numerical schemes employing Gegenbauer polynomials as basis polynomials. 27,30,32 In practice, we compute Gegenbauer polynomials using the following recurrence relation:…”

“…The necessary formulas to compute the two-fold integrals of Gegenbauer polynomials can be found in a previous work. 27 However, in typical PS and nodal DG methods employing Gegenbauer basis polynomials, the GIM is applied to approximate the needed definite integrals. To this end, the definite integration of some integrand function, say f (x), is approximated by integrating the Gegenbauer interpolant in its Lagrange form, say f n (x), and the sought definite integration approximations based on the GG interpolation nodes x i , i = 0, … , n, can be expressed in the following matrix-vector multiplication form: (…”

“…The current research paper objectives are highly motivated by the desire to extend the recent fully exponentially convergent numerical algorithms presented in previous works 1,27,28 using nodal discontinuous Galerkin (DG) methods to cover nonlinear PDE-based OCPs. In particular, we introduce "the linearization, integral, nodal discontinuous Gegenbauer-Galerkin (LINDG-G) method": a novel computational algorithm that draws its power from the integration of a novel linearization strategy with well-conditioned integral reformulations, optimal barycentric Gegenbauer quadratures, and the superior advantages possessed by the family of DG methods to furnish high-order, stable, and fully exponentially convergent numerical approximations of minimizers for OCPs governed by scalar PDEs of Burgers' type in one space dimension and exhibiting sufficiently smooth state and control variables.…”

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

“…[36][37][38][39][40][41][42][43][44][45][46][47][48] To avoid the ill conditioning of differential operators and the reduction in convergence rate for derivatives, an alternative direction to the aforementioned methods is to recast the partial differential equation into its integral formulation to take advantage of the well conditioning of integral operators and then discretize the latter using various discretization techniques. This useful strategy was applied successfully in the recent papers of Elgindy, 48,49 in which highly accurate numerical schemes were established to solve the second-order one-dimensional hyperbolic telegraph equation and parabolic distributed parameter system-based optimal control problems.…”

“…The current research paper objectives are highly motivated by the desire to extend the recent works of Elgindy 48,49 to solve Burgers' equation as one of the fundamental nonlinear partial differential equations that drew much attention over decades. In particular, we shall endeavor through this research paper to establish some novel computational algorithms suited to provide highly accurate, fully exponentially convergent, fast, and stable numerical solutions.…”