Core Concepts
The author presents novel spectral methods for shock capturing and tyger removal in computational fluid dynamics, aiming to converge strongly to the entropic weak solution under specific parameters.
Abstract
The content discusses the application of spectral methods in fluid dynamics, focusing on shock capturing and tyger removal. It introduces novel spectral relaxation and purging schemes to address Gibbs oscillations and tygers in approximations of nonlinear conservation laws. The study extends to various systems of hyperbolic conservation laws, showcasing numerical investigations and efficiency in shock capture. The comparison with traditional methods highlights the advantages of spectral schemes in accuracy and convergence rates.
Key points include:
- Introduction of spectral methods for solving fluid dynamics PDEs.
- Challenges faced by traditional methods with discontinuities like shocks.
- Novel spectral relaxation (SR) and purging (SP) schemes proposed for better approximations.
- Detailed numerical investigation on 1D inviscid Burgers equation and other systems.
- Comparison with low-order schemes regarding accuracy, convergence, and computational cost.
- Discussion on controlling high-frequency growth using different kernels.
- Importance of parameter selection for optimal performance of SR and SP schemes.
Overall, the content emphasizes the potential of spectral methods in improving shock capture efficiency while addressing issues like Gibbs oscillations and tygers.
Stats
For the 1D inviscid Burgers equation, it is shown that the novel SR and SP approximations converge strongly in L2 norm to the entropic weak solution under an appropriate choice of kernels and related parameters.
In panel (c), we see that the tygers have spread throughout the domain.
Quotes
"Spectral solutions use global functions converging to exact smooth solutions."
"High-order finite-element schemes perform well but are complex."