In　this　paper,　we　propose　a　local　discontinuous　Galerkin　(LDG)　method　for　the　Novikov　equation　that　contains　cubic　nonlinear　high-order　derivatives.　Flux　correction　techniques　are　used　to　ensure　the　stability　of　the　numerical　scheme.　The　H　1-norm　stability　of　the　general　solution　and　the　error　estimate　for　smooth　solutions　without　using　any　priori　assumptions　are　presented.　Numerical　examples　demonstrate　the　accuracy　and　capability　of　the　LDG　method　for　solving　the　Novikov　equation.