Convergence of a Finite Difference Scheme for the Camassa–Holm Equation

2006/01/01 by Helge Holden, Xavier Raynaud · 65 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Mathematical Physics Problems #Fractional Differential Equations Solutions #Mathematics #Camassa–Holm equation #Convergence (economics) #Initial value problem #Mathematical analysis #Finite difference method #Finite difference #Central differencing scheme #Boundary value problem #Scheme (mathematics) #Finite difference scheme #Energy method #Applied mathematics #Finite difference coefficient #Finite element method #Physics #Mixed finite element method #Integrable system

paper · doi:10.1137/040611975

published in SIAM Journal on Numerical Analysis 44(4), 1655-1680 (Society for Industrial and Applied Mathematics)

openalex publication_date 2006/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11

Abstract

We prove that a certain finite difference scheme converges to the weak solution of the Cauchy problem on a finite interval with periodic boundary conditions for the Camassa–Holm equation u_t-u_xxt+3uu_x-2u_xu_xx-uu_xxx=0 with initial data u|_t=0=u₀∈ H¹([0,1]). Here it is assumed that u₀-u₀''≥0, and in this case the solution is unique, globally defined, and energy preserving.

Citations

• A New Integrable Shallow Water Equation
• Classical solutions of the periodic Camassa—Holm equation
• Camassa–Holm, Korteweg–de Vries and related models for water waves
• Global Weak Solutions for a Shallow Water Equation
• On the Cauchy problem for the Camassa–Holm equation
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• Integral and integrable algorithms for a nonlinear shallow-water wave equation
• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• On the Blow-Up of Solutions of a Periodic Shallow Water Equation

Cited by

• GLOBAL CONSERVATIVE MULTIPEAKON SOLUTIONS OF THE CAMASSA–HOLM EQUATION
• A Convergent Finite Difference Scheme for the Camassa–Holm Equation with General H¹ Initial Data
• A convergent numerical scheme for the Camassa--Holm equation based on multipeakons
• Periodic conservative solutions of the Camassa–Holm equation
• Camassa–Holm equation [wikipedia]