| With the rapid development of science and technology,many partial differential equations become popular models to describe and resolve more and more physical,chemical and geological problems.Generally,it is difficult to derive the exact so-lutions for partial differential equations(PDEs)and ordinary differential equations(ODEs).Many authors focus on designing numerical methods.It is well-known that many continuous systems have some essential properties of the original system.Thus,we aim to construct new methods for PDEs to preserve as many as properties of the continuous system as possible.The mass conservation law and the energy conservation law are not only im-portant properties,but also become standards to judge the effectiveness of numerical methods.Thus,construction mass preserving and energy preserving methods have aroused great interesting for many researches.We note that the relevant works are mainly based on the finite difference methods and the spectral collation methods.In this paper,we specially develop an energy preserving method based on the Galerkin spectral element method.We prove that the semi-discrete system of the coupled non-linear Schrodinger equations holds the Hamiltonian structure based on the linear el-ement.Later,we rigorously prove that the fully discrete method admits the discrete mass conservation law and the energy conservation law.Moreover,we utilize the conservative properties,the classical project theory to derive the convergent rate,i.e.,O(?2 + h2)in the discrete L2-norm.Finally,comprehensive numerical experiments are given to illustrate the superiority of the proposed energy-preserving method.The symplectic conservation law is an important property for Hamiltonian sys-tem.In 1984,Feng proposed the symplectic methods which were robust,efficient and very accurate in preserving the long-time behavior of solutions of Hamiltonian systems.Subsequently,extensive numerical studies about symplectic methods have been obtained in the literature.We note that the most existing symplectic methods for Hamiltonian PDEs are based on the finite difference method,the wavelet colloca-tion method,the Fourier pseudospectral method.In contrast,the study of symplectic methods constructed from finite element method(FEM)is still in its infancy.There are a few symplectic methods for 2D Hamiltonian equations in the literature.Further-more,we note that most of the symplectic methods are completely implicit,which greatly suppress the efficiency.Thus,we specially construct a symplectic preserving and fully implicit method for the 2D nonlinear Schrodinger equation(2D NLS)by the Galerkin finite element method.Then we transform the fully implicit method into the equivalent explicit method by the FFT technique to increase computation efficien-cy.We rigorously prove our method is an unconditionally linear stable and conserve the discrete mass and symplectic conservation laws.Numerical experiments confirm that the proposed method provides accurate solutions in long-term computations and conserves corresponding conservative properties.Structure-preserving methods are mainly applied to solve the conservative Hamiltonian PDEs,there are few numerical researches on general PDEs with energy dissipation properties.General PDEs with energy dissipation properties could contain more dynamic systems and be applied to more extensive fields,such as Birkhoffi-an dynamics,oscillatory systems and conformal Hamiltonian systems etc.Here,we consider the Cahn-Hilliard(CH)equation.The incompressible multi-phase flow problem is a hot problem in the multi-material hybrid fluid mechanics system.The Cahn-Hilliard(CH)equation is a widely used phase-field models which is popular method to deal with the free interface prob-lem of multi-phase incompressible flows.So far,extensive mathematical studies have been carried out for the CH equation in the literature.We note that most existing methods and their stabilities discussed in the literature are based on the global energy dissipation property.In this paper,we derive that CH equation has a local energy dis-sipation property which is independent of boundary condition.We specially construct three novel numerical methods to preserve discrete analogues of the local energy dis-sipation law for the 2D CH equation through the concatenation of basic algorithms,including the implicit midpoint method,the leap-frog method and the discrete vari-ational derivative method for the nonlinear term.The corresponding discrete local energy dissipation laws are rigorously proved by the discrete Leibnitz rules.While imposed of the periodic boundary condition or the homogeneous boundary condition,the proposed methods thereby possess the global energy dissipation law as well as mass conservation law.Numerical experiments are given to illustrate the superiority of the proposed methods.For the CH equation,it is well-known that explicit methods are much easy for implementation,but they usually suffer from severe time step restrictions and do not obey the global energy dissipation law.Whereas,fully implicit methods usually are not efficient in practice,but usually unconditionally energy stable.Consequently,linear-implicit methods are the compromising approaches to solve the CH equation.Many works have been achieved including the convex splitting skills proposed by Elliott et al.and the new technique called“invariant energy quadratization".We note that most existing methods and their stabilities discussed in the literature are based on the global energy dissipation laws which are dependent on the suitable boundary conditions.Thus,we specially construct four linear-implicit local energy dissipation law methods by the newly induced invariant energy quadratization method,for the 2D CH equation with a variable mobility.We rigorously prove that the four new methods preserve the discrete local energy dissipation law in any time-space region.With periodic boundary conditions,the proposed methods are proven to conserve the discrete total mass conservation and global energy dissipation laws. |
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