The Direct Discontinuous Galerkin Method For Allen-Cahn Equations

In this paper,we consider the direct discontinuous Galerkin method for the Allen-Cahn equation in one and two dimensionals.For the time discreatization,we consider the forward and backward Euler method.With a suitable linearization method,a semi implicit scheme is also proposed.The numerical results show that the explicit scheme requires relatively small time step for the stablity.A number of numerical examples are presented to verity the stability and efficient of the proposed direct discontinuous Galerkin methods.