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：谱方法可提供非线性偏微分方程（特别是流体动力学方程）在Galerkin截断近似下的数值解。当解存在间断（如激波）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。针对一维无粘Burgers方程的谱近似，非线性波共振会在远离激波的流动充分解析区域引发tyger现象。近期Besse（待发表）提出了新型谱松弛（SR）和谱净化（SP）格式，用于消除非线性守恒律谱近似中的tyger和吉布斯振荡。研究表明，通过适当选择核函数及相关参数，一维无粘Burgers方程的SR和SP近似解能在L2范数意义下强收敛到熵弱解。本文首先对SR和SP格式应用于一维无粘Burgers方程进行详细数值研究，报告其在激波捕捉和tyger消除方面的效率。随后将研究拓展至非线性双曲守恒律系统——包括浅水方程的2×2系统和标准一维可压缩欧拉方程的3×3系统。针对后者，我们利用切比雪夫多项式将SR方法的实现推广至非周期问题。最后转向三维轴对称有界不可压缩欧拉方程的一维壁面近似中的奇异流动：为确定解的大爆裂时间，我们将纯拟谱方法获得的解析性带宽度衰减与新型谱松弛格式改进后的估计值进行对比。