Kolluru, SSV and Besse, N and Pandit, R (2024) Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics. In: Journal of Computational Physics, 519 .
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Abstract
Spectral methods yield numerical solutions of the Galerkin-truncated versions of nonlinear partial differential equations (PDEs) 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 inviscid Burgers equation in one-dimension (1D), nonlinear wave resonances lead to the formation of localised oscillatory structures called tygers in well-resolved areas of the flow, far from the shock. Recently, the author of 1 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 in 1 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 2�2 system of the shallow water equations and the standard 3�3 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 2. 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 (PPS), with the improved estimate obtained using the novel spectral relaxation scheme. © 2024 Elsevier Inc.
| Item Type: | Journal Article |
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| Publication: | Journal of Computational Physics |
| Publisher: | Academic Press Inc. |
| Additional Information: | The copyright for this article belongs to the authors. |
| Keywords: | Chebyshev polynomials; Galerkin methods; Nonlinear equations; Oscillating flow; Polynomials, Burger's equations; Computational fluid; Fluid-dynamics; Gibbs oscillations; Numerical solution; One dimension; Shock-capturing; Spectral approximations; Spectral methods; Spectral relaxations, Euler equations |
| Department/Centre: | Division of Physical & Mathematical Sciences > Physics |
| Date Deposited: | 01 Oct 2024 11:07 |
| Last Modified: | 01 Oct 2024 11:07 |
| URI: | http://eprints.iisc.ac.in/id/eprint/86317 |
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