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On error estimates for Galerkin finite element methods for the Camassa-Holm equation

Antonopoulos, D. C. ; Dougalis, V. A. ; Mitsotakis, D. E.

Arxiv ID: 1805.10744

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  • Nhan đề:
    On error estimates for Galerkin finite element methods for the Camassa-Holm equation
  • Tác giả: Antonopoulos, D. C. ; Dougalis, V. A. ; Mitsotakis, D. E.
  • Chủ đề: Mathematics - Numerical Analysis ; 65m60, 35q53
  • Mô tả: We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $L^{2}$-error estimates for the semidiscrete approximation. We also consider an initial-boundary-value problem on a finite interval for the system form of CH and analyze the convergence of its standard Galerkin semidiscretization. Using the fourth-order accurate, explicit, "classical" Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the `peakon' type.
  • Số nhận dạng: Arxiv ID: 1805.10744

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