The Use of Chebyshev Polynomials in the Space–Time Least-Squares Spectral Element Method

Chebyshev polynomials of the first kind are employed in a space–time least-squares spectral element formulation applied to linear and nonlinear hyperbolic scalar equations. No stabilization techniques are required to render a stable, high order accurate scheme. In parts of the domain where the underlying exact solution is smooth, the scheme exhibits exponential convergence with polynomial enrichment, whereas in parts of the domain where the underlying exact solution contains discontinuities the solution displays a Gibbs-like behavior. An edge detection method is employed to determine the position of the discontinuity. Piecewise reconstruction of the numerical solution retrieves a monotone solution. Numerical results will be given in which the capabilities of the space–time formulation to capture discontinuities will be demonstrated.