Conservative Local Discontinuous Galerkin method for the fractional Klein-Gordon-Schrödinger system with generalized Yukawa interaction

The formulation of the Local Discontinuous Galerkin (LDG) method applied to the space fractional Klein-Gordon-Schrödinger system with generalized interaction is presented. By considering its primal formulation and taking advantage of the symmetry of the bilinear form associated to the discretization of the Riesz differential operator, conservation of discrete analogues of the mass and the energy can be demonstrated for the semi-discrete problem and for the fully discrete problem using, as time marching scheme, a combination of the modified Crank-Nicolson method for the fractional nonlinear Schrödinger equation and the Newmark method for the nonlinear Klein-Gordon equation. Boundedness of the numerical solution in the L2 norm is derived from the conservation properties of the fully discrete method. A series of numerical experiments with high order approximations illustrates our conservation results and shows that optimal rates of convergence can be also achieved.