Conservative or dissipative quasi-interpolation method for evolutionary partial differential equations

Based on quasi-interpolation, we propose a new meshless conservative or dissipative method for nonlinear time-dependent partial differential equations. Using the method of lines, we first discretize the equation in space with the quasi-interpolation method, then employ the average vector field method in time discretization to derive the final numerical scheme. The method not only inherits the conservation or dissipation property of the equation but also has the meshless feature since we use the nonuniform grids in spatial discretization. Several numerical examples are presented to demonstrate the effectiveness of the proposed method.