An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg-De Vries equation
Provides an energy-conserving numerical method for solving the KdV equation, which is important for long-time simulations in computational physics.
An energy-conserving ultra-weak discontinuous Galerkin method is proposed for the generalized KdV equation, achieving optimal convergence order k+1 for semi-discrete schemes without convection and superior long-time simulation performance over existing DG methods.
We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) method for the generalized Korteweg-De Vries(KdV) equation in one dimension. Optimal a priori error estimate of order $k + 1$ is obtained for the semi-discrete scheme for the KdV equation without convection term on general nonuniform meshes when polynomials of degree $k\ge 2$ is used. We also numerically observed optimal convergence of the method for the KdV equation with linear or nonlinear convection terms. It is numerically observed for the new method to have a superior performance for long-time simulations over existing DG methods.