High-order integral nodal discontinuous Gegenbauer-Galerkin method for solving viscous Burgers' equation

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

“…(ii) All equality constraints are formulated in terms of integral operators that are widely popular for being "well-conditioned operators," and "their well-conditioning is essentially unaffected for increasing number of points"; see other works. 28,[32][33][34] Therefore, the approximate auxiliary state and control variables can be recovered with nearly full machine precision without using any preconditioners. On the other hand, direct discretization of the dynamical system Equation (1a) in its strong differential form using DG methods typically lead to ill-conditioned nonlinear algebraic constraint system equations that generally suffer from degradation of precision, as the condition number of the pth-order numerical differential operator grows like N 2p .…”

“…The rest of the proof is established by analyzing the truncation error of Equation (B7a), which typically follows the proof in a previous work (Theorem 7.1). 28 Theorem 1 shows that the semidiscretization truncation error decays exponentially fast as N and M → ∞ if the constant C < 2; it can be shown further that this condition is satisfied when…”

“…In fact, Theorem 1 manifests that while holding the value of M fixed, the truncation error upper bound of the semidiscretization roughly increases for increasing values of |c| and decreasing values of "since the constant c 1,j is proportional with | |, which is linearly proportional to the magnitude of the parameter |c| but inversely proportional to the viscosity parameter "; see previous work. 28 On the other hand, Theorems 2 and 3 reveal an opposite story: the truncation error upper bound of the full discretization roughly increases for increasing values of since the parameters 1 and 2 are linearly dependent on . It is interesting here to find that the potential rise of truncation error in the semidiscretization stage corresponding to increasing values of |c| and decreasing values of can be avoided by simply increasing the values of M without any change to the number of collocation points, thanks to the development of optimal Gegenbauer matrices of rectangular forms.…”

“…(iii) The work of Doha 36 in approximating the solution of linear PDEs in one dimension shows that higher-order approximations better than those obtained from Chebyshev and Legendre polynomials can be obtained from Gegenbauer polynomial expansions for small and negative values of . (iv) The recent works of Elgindy et al 1,28,32,[37][38][39] show that Gegenbauer polynomial methods are very effective in the solutions of BVPs, integral equations, integro-differential equations, fractional differential equations, and OCPs. Moreover, the reported results illustrate that some members of the Gegenbauer family of polynomials converge to the solutions of the aforementioned problems faster than Chebyshev and Legendre polynomials for the small/medium range of the number of spectral expansion terms.…”

“…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.…”