Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics

TL;DR

The paper addresses Gibbs oscillations and tygers that plague spectral methods for nonlinear hyperbolic PDEs by introducing spectral relaxation (SR) and spectral purging (SP) schemes. These methods use kernel-based relaxation and discrete-time purging to inject controlled dissipation while preserving high-order accuracy in smooth regions, and they are extended from periodic Burgers to 2×2 shallow-water and 3×3 Euler systems with nonperiodic Chebyshev implementations and CCM boundary handling. The authors demonstrate strong $L^2$ convergence to entropic weak solutions for Burgers, robust shock-capturing across multiple test problems, and an extended ability to track finite-time singularities via the analyticity-strip method, compared to standard PPS and SVV approaches. Overall, SR and SP offer a flexible, high-order framework for shock-capturing in spectral methods, enabling reliable simulations of complex flows while maintaining spectral convergence in smooth regions and compatibility with nonperiodic geometries. These advances have practical implications for efficiently solving high-Reynolds-number flows with discontinuities in physics-informed simulations and rigorous singularity tracking.

Abstract

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.

Figures (17)

• Figure 1: Plots vs. $x$ of the velocity field $u(x,t)$ at increasing times from panels $(a)-(c)$. The exact solution (black line) and the $2/3$-dealiased PPS approximation on $N_x=615$ points (grey line) are superposed for comparison. We see the development of Gibbs oscillations near the shock and tygers in the smooth parts of the field which experience a positive strain. In panel $(c)$, we see that the tygers have spread throughout the domain.
• Figure 2: Plots vs. $x$ of the velocity field $u(x,t)$ obtained by using $(a)$ spectral relaxation (SR) and $(b)$ spectral purging (SP) schemes with the Fejér--Korovkin kernel. The value of $\alpha$ used is given in the legend; we set $\gamma=0.99$. The plots are shown for time $t=0.2$ after the shock formation at $t_* \simeq 0.159$. We compare these approximations with the exact solution (black line) and the $2/3$-dealiased PPS solution (light grey line). The insets in each panel are used to zoom into the region of the shock.
• Figure 3: Plots vs. $x$ of the velocity field $u$ approximated using the best SR-FeKo (orange crosses) and SP-FeKo (blue circles) schemes. The parameter sets are, for SR-FeKo: $(\alpha=0.7, \gamma=0.99)$ and for SP-FeKo: $(\alpha=0.65, \gamma=0.99)$. The exact solution (black line) and the $2/3$-dealiased PPS solution (grey line) are plotted for comparison. We use a spatial resolution of $N_x = 615$ for all the runs shown here.
• Figure 4: Plot vs. $\alpha$ of the order of convergence calculated from $L^\infty$-errors of the SR-FeKo approximation with $(\alpha,\gamma=0.99)$ at early time $t=0.07$ when the solution is still smooth.
• Figure 5: Plots vs. $x$ of the velocity field $u$ obtained by using SR-FeKo for $(\alpha=0.7,\gamma=0.99)$ (blue crosses) and SR-DlVP (with the de La Vallée Poussin filter) for $(\alpha=0.89,\gamma=0.9)$ (orange plusses) at different representative times in panels $(a)$-$(c)$ before and after the shock formation. We compare these approximations with the exact solution (black line) and the $2/3$-dealiased PPS solution (light grey line). The zoomed insets in each panel show Gibbs-like oscillations in SR-DlVP approximation. The spatial resolution is $N_x=615$.