A Convergent Finite Difference Scheme for the Camassa–Holm Equation with General H¹ Initial Data

2008/01/01 by Giuseppe Maria Coclite, Kenneth H. Karlsen, Nils Henrik Risebro · 51 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Mathematical Physics Problems #Algebraic structures and combinatorial models #Mathematics #Camassa–Holm equation #Dissipative system #Scheme (mathematics) #Central differencing scheme #Finite difference #Applied mathematics #Finite difference scheme #Finite difference method #Mathematical analysis #Finite difference coefficient #Finite element method #Mixed finite element method #Integrable system #Physics

paper · open access · doi:10.1137/060673242

published in SIAM Journal on Numerical Analysis 46(3), 1554-1579 (Society for Industrial and Applied Mathematics)

openalex publication_date 2008/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We suggest a finite dfference scheme for the Camassa–Holm equation that can handle general H¹ initial data. The form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality. We prove that the difference scheme converges strongly in H¹ towards a dissipative weak solution of the Camassa–Holm equation.

Citations

• A New Integrable Shallow Water Equation
• A note on well-posedness for Camassa–Holm equation
• Numerical simulation of Camassa–Holm peakons by adaptive upwinding
• GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION
• Camassa–Holm, Korteweg–de Vries and related models for water waves
• ON THE UNIQUENESS AND LARGE TIME BEHAVIOR OF THE WEAK SOLUTIONS TO A SHALLOW WATER EQUATION
• Global Conservative Solutions of the Camassa–Holm Equation
• Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod
• Global Weak Solutions for a Shallow Water Equation
• On the Cauchy problem for the Camassa–Holm equation
• On the scattering problem for the Camassa-Holm equation
• A shallow water equation as a geodesic flow on the Bott-Virasoro group
• Integral and integrable algorithms for a nonlinear shallow-water wave equation
• Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View
• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Wave breaking for nonlinear nonlocal shallow water equations
• Geodesic flow on the diffeomorphism group of the circle
• Convergence of a Finite Difference Scheme for the Camassa–Holm Equation
• Multipeakons and the Classical Moment Problem

Cited by

• GLOBAL CONSERVATIVE MULTIPEAKON SOLUTIONS OF THE CAMASSA–HOLM EQUATION
• Camassa–Holm equation [wikipedia]