A fast solver of Legendre-Laguerre spectral element method for the Camassa-Holm equation
- PDF / 1,284,525 Bytes
- 23 Pages / 439.642 x 666.49 pts Page_size
- 86 Downloads / 181 Views
A fast solver of Legendre-Laguerre spectral element method for the Camassa-Holm equation Xuhong Yu1 · Xueqin Ye1 · Zhongqing Wang1 Received: 16 May 2020 / Accepted: 7 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract An efficient and accurate Legendre-Laguerre spectral element method for solving the Camassa-Holm equation on the half line is proposed. The spectral element method has the flexibility for arbitrary h and p adaptivity. Two kinds of Sobolev orthogonal basis functions corresponding to each subinterval are constructed, which reduces the non-zero entries of linear systems and computational cost. Numerical experiments illustrate the effectiveness and accuracy of the suggested approach. Keywords Legendre-Laguerre spectral element method · The Camassa-Holm equation · Diagonalization technique · Numerical results Mathematics Subject Classification (2010) 65M70 · 35Q35 · 33C45
1 Introduction The spectral method, with its high-order accuracy, is becoming an increasingly popular numerical method (cf. [3, 6, 12, 22] and the references therein). Theoretically, higher approximation orders result in smaller numerical errors. However, it is not convenient to use very large modes in actual computation of spectral method. A reasonable way is to use the spectral element method to approximate the problems under consideration. The spectral element method has the flexibility for arbitrary h and p adaptivity, and has greatly promoted the development of the classical spectral method; see, e.g., [20, 23, 26].
Zhongqing Wang
[email protected] 1
University of Shanghai for Science and Technology, Shanghai, 200093, China
Numerical Algorithms
In this paper, we consider numerical simulations to the Camassa-Holm (CH) equation on the half line using the Legendre-Laguerre spectral element method: ⎧ u − uxxt + 3uux = 2ux uxx + uuxxx , t ∈ (0, T ], x ∈ (0, +∞), ⎪ ⎨ t x ∈ [0, +∞), u(x, 0) = u0 (x), (1.1) ⎪ ⎩ t ∈ [0, T ], u(0, t) = ux (0, t) = 0, where u0 (x) is assumed to be fast decaying as x → ∞ and satisfying u0 (0) = ∂x u0 (0) = 0. The CH equation was introduced as a model for the propagation of unidirectional shallow water waves by Camassa and Holm [5], with u(x, t) representing the water’s free surface in non-dimensional variables. It supports an infinite number of smooth waves (called solitons) and non-smooth solitary wave solutions (called peakons), and possesses an infinite number of conserved integrals. Because of these remarkable properties, a series of works with respect to the theoretical analysis of the CH equation have been done in the past decades [11, 14, 19]. Some related numerical methods have also been developed for the CH equation, mainly including the finite difference method [7, 8, 15], finite volume method [2], pseudospectral method [16– 18], local discontinuous Galerkin method [25], operator splitting methods [4], and multisymplectic methods [9, 10, 27]. To the best of our knowledge, there are very few literatures concerning spectral element methods for the CH eq
Data Loading...
