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We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) methodfor the generalized Korteweg-De Vries(KdV) equation in one dimension. Optimal apriori error estimate of order $k + 1$ is obtained for the semi-discrete schemefor the KdV equation without convection term on general nonuniform meshes whenpolynomials of degree $k\ge 2$ is used. We also numerically observed optimalconvergence of the method for the KdV equation with linear or nonlinearconvection terms.  It is numerically observed for the new method to have a superior performancefor long-time simulations over existing DG methods.

An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg-De Vries equation