A SPECTRAL DAMPING METHOD FOR NONLINEAR CONSERVATION LAWS
A SPECTRAL DAMPING METHOD FOR NONLINEAR CONSERVATION LAWS
Lin Sang 1Juan Cheng 1Chi-Wang Shu2
作者信息
- 1. Capital Normal University
- 2. Brown University
- 折叠
摘要
Although spectral methods provide highly accurate approximations to smooth solutions, they typically perform poorly for nonlinear conservation laws when shock discontinuities develop. In this work, motivated bythe numerical damping of discontinuous Galerkin methods [J. Lu, Y. Liu andC.-W. Shu, SIAM J. Numer. Anal., 59 (2021), pp. 1299-1324], we introducea novel numerical damping term to stabilize the Fourier spectral method fornonlinear conservation laws. With this damping, the Fourier coefficients ofthe numerical solution decay rapidly. Consequently, we rigorously prove thatthe spectrally damped approximation is uniformly bounded, has bounded totalvariation, and satisfies the entropy condition in the limit, thereby guaranteeingconvergence to the unique entropy solution. Furthermore, we demonstrate thatthe method retains spectral accuracy for smooth solutions. Compared withthe classical spectral viscosity method [E. Tadmor, SIAM J. Numer. Anal.,26 (1989), pp. 30-44], the proposed method shows improved performance inour numerical experiments.
Abstract
Although spectral methods provide highly accurate approximations to smooth solutions, they typically perform poorly for nonlinear conservation laws when shock discontinuities develop. In this work, motivated bythe numerical damping of discontinuous Galerkin methods [J. Lu, Y. Liu andC.-W. Shu, SIAM J. Numer. Anal., 59 (2021), pp. 1299-1324], we introducea novel numerical damping term to stabilize the Fourier spectral method fornonlinear conservation laws. With this damping, the Fourier coefficients ofthe numerical solution decay rapidly. Consequently, we rigorously prove thatthe spectrally damped approximation is uniformly bounded, has bounded totalvariation, and satisfies the entropy condition in the limit, thereby guaranteeingconvergence to the unique entropy solution. Furthermore, we demonstrate thatthe method retains spectral accuracy for smooth solutions. Compared withthe classical spectral viscosity method [E. Tadmor, SIAM J. Numer. Anal.,26 (1989), pp. 30-44], the proposed method shows improved performance inour numerical experiments.关键词
Conservation laws/spectral damping method/total variation/entropy solution/convergence.Key words
Conservation laws/spectral damping method/total variation/entropy solution/convergence.引用本文复制引用
Lin Sang,Juan Cheng,Chi-Wang Shu.A SPECTRAL DAMPING METHOD FOR NONLINEAR CONSERVATION LAWS[EB/OL].(2026-10-12)[2026-10-13].https://chinaxiv.org/abs/202610.00069.学科分类
数学
