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New conservative finite volume element schemes for the modified Korteweg-de Vries equation. (English) Zbl 1359.65169

The authors consider the numerical approximation of the one-dimensional modified Korteweg-de Vries equation \[ u_t+\varepsilon u^2 + \mu u_{xxx} = 0. \] Conservative finite volume element schemes are proposed and compared, especially with regard to their accuracy and conservative properties. The schemes are constructed based on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation.
The paper is organized as follows. Section 1 is an introduction. Preliminaries and notifications are given in Section 2. The schemes and corresponding properties are presented in Section 3. Section 4 is devoted to the linear stability analysis of three schemes. Numerical experiments are given in Section 5. Some conclusions are given in the last Section 6.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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