zbMATH — the first resource for mathematics

Finite difference approximate solutions for the Rosenau equation. (English) Zbl 0904.65093
Numerical solutions of a finite difference scheme are discussed for the KdV-like Rosenau equation \[ u_t+ u_{xxxxt} +uu_x +u_x=0, \quad (x,t)\in (0,1)\times (0,T] \] with an initial condition \[ u(x,0)= u_0(x), \quad x\in (0,1) \] and boundary conditions \[ u(0,t)= u(1,t)=0, \quad u_{xx} (0,t)= u_{xx} (1,t) =0. \] This equation was modelled by P. Rosenau [Dynamics of dense discrete systems, Prog. Theoretical Phys. 79, 1028-1042 (1988)] in order to describe the dynamics of dense discrete systems.
Existence and uniqueness of the solution for the scheme are shown by using the Brouwer fixed point theorem. An a priori bound and convergence of order \(O(h^2 +k^2)\) as well as conservation of energy of the finite difference approximate solutions are discussed with numerical examples.
Reviewer: S.K.Chung (Seoul)

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI
[1] Browder F. E., Esistence and uniqueness theorems for solutions of nonlinear boundary value problems, Applications of nonlinear partial differential equations (1965)
[2] DOI: 10.1080/00036819408840267 · Zbl 0830.65097
[3] DOI: 10.1016/0362-546X(93)90179-V · Zbl 0811.35142
[4] DOI: 10.1143/PTP.79.1028
[5] DOI: 10.1088/0031-8949/34/6B/020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.