An adaptive spectral least-squares scheme for the Burgers equation

Abstract

A least-squares spectral collocation method for the one-dimensional inviscid Burgers equation is proposed. This model problem shows the stability and high accuracy of these schemes for nonlinear hyperbolic scalar equations. Here we make use of a least-squares spectral approach which was already used in an earlier paper for discontinuous and singular perturbation problems (Heinrichs, J. Comput. Appl. Math. 157:329-345, 2003). The domain is decomposed in subintervals where continuity is enforced at the interfaces. Equal order polynomials are used on all subdomains. For the spectral collocation scheme Chebyshev polynomials are employed which allow the efficient implementation with Fast Fourier Transforms (FFTs). The collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The scheme exhibits exponential convergence where the exact solution is smooth. In parts of the domain where the solution contains discontinuities (shocks) the spectral solution displays a Gibbs-like behavior. Here this is overcome by some suitable exponential filtering at each time level. Here we observe that by over-collocation the results remain stable also for increasing filter parameters and also without filtering. Furthermore by an adaptive grid refinement we were able to locate the precise position of the discontinuity. Numerical simulations confirm the high accuracy of our spectral least-squares scheme.

An Improved Adaptive hp Algorithm using Pseudospectral Methods for Optimal Control

“…(13) subject to the algebraic constraints of Eqs. (14)- (17). Just as in the continuoustime optimal control problem of Eqs.…”

Effective numerical technique applied for Burgers' equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms

“…Numerous numerical algorithms were exploited to numerically solve NLEEs especially Burgers' equation to achieve minimized errors with respect to analytical solutions. Finite difference and other modifications, 18–24 finite element and B‐spline finite element, 25–27 spectral least squares method, 28–30 variational iteration method, 31,32 Adomian–Pade technique, 33 homotopy analysis, 34 and automatic differentiation method 35 are examples of such numerical techniques. Moreover, miscellaneous numerical techniques have been employed either for Burgers' equation or other engineering applications such as boundary element techniques for cavitation of hydrofoils, 36,37 step cubic polynomial, 38 technique of modified diffusion coefficient for studying convection diffusion equation, 39 and differential quadrature for functionally graded nanobeams 40,41 .…”

The pointwise estimates of a conservative difference scheme for Burgers' equation

“…Up to now, there are already a variety of analytical methods to solve (1.1)–(1.3), such as Cole–Hopf transformation [1, 2], variational iteration method [3], Adomian's decomposition method [4], and so on. For the numerical methods, there are finite difference method [5, 6], finite element method [7, 8], multiquadric RBF method [9], Haar wavelets method [10], adaptive spectral least‐squares method [11], high order splitting method [12, 13], Haar wavelet quasilinearization approach [14], weighted average differential quadrature method [15], exponential modified cubic B‐spline differential quadrature method [16], meshfree methods [17], and so on. However, most of the numerical methods focus on the L 2 ‐error convergence or lack of numerical analysis, and few works concentrate on the L ∞ ‐error convergence, especially for the nonlinear implicit schemes.…”