Convergence of an efficient and compact finite difference scheme for the Klein-Gordon-Zakharov equation

A compact and semi-explicit finite difference scheme is proposed and analyzed for the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is decoupled and linearized in practical computation, i.e., at each time step only two tri-diagonal systems of linear algebraic equations need to be solved by Thomas algorithm. So the new scheme is more efficient and more accurate than the classical finite difference schemes. Unique solvability of the difference solution is proved by using the energy method. Besides the standard energy method, in order to overcome the difficulty in obtaining the a priori estimate, an induction argument is introduced to prove that the new scheme is convergent for u(x,t) in the discrete H1-norm, and respectively for m(x,t) in the discrete L2-norm, at the order of O(τ2+h4) with time step τ and mesh size h. Numerical results are reported to verify the theoretical analysis.