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.

翻译：谱方法用于求解非线性偏微分方程（尤其涉及流体动力学）的伽辽金截断近似数值解。当存在激波等间断时，谱近似会在间断附近产生吉布斯振荡，导致数值解迅速偏离真实解。针对一维无黏伯格斯方程的谱近似，非线性波共振会在远离激波的流动良好解析区形成“虎斑”。近期，Besse（待发表）提出了新型谱松弛（SR）与谱清除（SP）方案，用于消除非线性守恒律谱近似中的“虎斑”与吉布斯振荡。研究表明，在一维无黏伯格斯方程中，通过适当选择核函数及相关参数，新型SR与SP近似解在L2范数下强收敛于熵弱解。本文对SR与SP方案应用于一维无黏伯格斯方程进行详细数值研究，报告其激波捕捉效率及“虎斑”消除效果。随后将研究扩展至非线性双曲守恒律方程组——如浅水方程的2×2系统及一维可压缩欧拉方程的标准3×3系统。针对后者，利用切比雪夫多项式将SR方法推广至非周期问题。进而研究三维轴对称壁面不可压缩欧拉方程的一维壁面近似奇异流动：为确定解爆破时间，对比纯伪谱法获得的解析带宽度衰减与采用新型谱松弛方案得到的改进估计。