  Spectral methods yield numerical solutions of the Galerkin-truncated versions
of nonlinear partial differential equations involved especially in fluid
dynamics. In the presence of discontinuities, such as shocks, spectral
approximations develop Gibbs oscillations near the discontinuity. This causes
the numerical solution to deviate quickly from the true solution. For spectral
approximations of the 1D inviscid Burgers equation, nonlinear wave resonances
lead to the formation of tygers in well-resolved areas of the flow, far from
the shock. Recently, Besse(to be published) has proposed novel spectral
relaxation (SR) and spectral purging (SP) schemes for the removal of tygers and
Gibbs oscillations in spectral approximations of nonlinear conservation laws.
For the 1D inviscid Burgers equation, it is shown that the novel SR and SP
approximations of the solution converge strongly in L2 norm to the entropic
weak solution, under an appropriate choice of kernels and related parameters.
In this work, we carry out a detailed numerical investigation of SR and SP
schemes when applied to the 1D inviscid Burgers equation and report the
efficiency of shock capture and the removal of tygers. We then extend our study
to systems of nonlinear hyperbolic conservation laws - such as the 2x2 system
of the shallow water equations and the standard 3x3 system of 1D compressible
Euler equations. For the latter, we generalise the implementation of SR methods
to non-periodic problems using Chebyshev polynomials. We then turn to singular
flow in the 1D wall approximation of the 3D-axisymmetric wall-bounded
incompressible Euler equation. Here, in order to determine the blowup time of
the solution, we compare the decay of the width of the analyticity strip,
obtained from the pure pseudospectral method, with the improved estimate
obtained using the novel spectral relaxation scheme.
