Numerical dynamics for discrete nonlinear damping Korteweg-de Vries equations

We study the numerical scheme of both solution and attractor for the time -space discrete nonlinear damping Korteweg-de Vries (KdV) equation, which is neither conservative nor coercive. First, we establish a new Taylor expansion as well as a global attractor for the KdV lattice system. Second, we prove the unique existence of numerical solution as well as numerical attractor for the discrete -time KdV lattice system via the implicit Euler scheme. Third, we estimate the discretization error and interpolation error between continuous -time and discretetime solutions, and then establish the upper semi -convergence from numerical attractors to the global attractor as the time -size tends to zero. Fourth, we establish the finitely dimensional approximation of numerical attractors. Finally, we establish the upper bound as well as the lower semi -convergence of numerical attractors with respect to the external force and the damping constant.