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We consider an initial boundary-value problem for the regularized long-wave equation. A three level one-parameter family of conservative difference schemes is studied. Two level schemes are used to find the values of the unknown functions on the first level. A numerical method for selection of artificial boundary conditions is proposed. It is proved that the finite difference scheme converges at rate O. 2 C h 2 / when an exact solution belongs to the Sobolev space W 3 2 . and the initial boundary-value problem with the conditions u.a; t/ D u.b; t / D 0; t 2 OE0; T /; u.x; 0/ D u 0 .x/; x 2 OEa; b; (1.2)can be considered in the domain Q T´. a; b/ .0; T /. The numerical solution of the RLW equation has been the subject of many papers (see e.g. [1, 4-10, 12, 18-20, 22]).The first finite-difference scheme with accuracy O. Ch 2 / is given by Peregrine [16]. In [9, 10], two and three level schemes are proposed and their convergence is proved.The two level scheme in [5] is conservative but passage from one layer to another requires iterations.In [22] the three level difference scheme is presented for problem (1.1), (1.2). The scheme is conservative, but, for the first layer, it is nonlinear with respect to the values of an unknown function. Convergence at rate O.h 2 C 2 / is proved under the condition that an exact solution belongs to C 4;3 .A three level scheme is considered in [18], showing convergence at rate O.h 2 C 2 / when an exact solution belongs to C 5 . Stability is proved for a sufficiently small mesh step. No way of calculation of the unknown function on the first level is suggested.A linearized implicit difference scheme with truncation error O. C h 2 / is presented in [12].A three level ten-point multisymplectic scheme is derived in [4]. In the present paper, a three level one-parameter family of difference schemes is constructed on a 9-point stencil. It is proved that the finite difference scheme converges at rate O. 2 C h 2 / when an exact solution belongs to the Sobolev space W 3 2 .Q T /. Steklov averaging operators are used for error estimation. In the present paper, a numerical method for selection of artificial boundary conditions is given, since the Dirichlet boundary conditions are zeros only within certain accuracy. A certain value of the free parameter of the scheme is also given, which, along with artificial boundary conditions, ensures good accuracy of an approximate solution.

Dedicated to Raytcho Lazarov on the occasion of his 60th birthday.Abstract -A boundary-value problem with a nonlocal integral condition is considered for a two-dimensional elliptic equation with constant coefficients and a mixed derivative. The existence and uniqueness of a weak solution of this problem are proved in a weighted Sobolev space. A difference scheme is constructed using the Steklov averaging operators. It is proved that the difference scheme converges in discrete W