Multi-Symplectic Methods For A Kind Of Hamiltonian Partial Differential Equation

In this paper we consider the multi-synipletic RKN method for the linear Boussi-nesq equationand the beam equationFirstly, we have derived (1) and (2) into the Hamiltonian formwhere, z∈R~n,S : R~n→R is some smooth function, and K, L arc two skew-symmetric constant n×n matrices. Next, we apply an r-stage RKN method for the temporal discretization and an s-stage RKN method for the spatial discretization, and get the fully-discrete schemes. Then, we proved that these schemes are multisymplectic. We also show two discrete multi-symplectic conservation laws, which corresponding to the linear Boussinesq equation and the beam equation, respectively. Theorem 1 If the above RKN-RKN discrete scheme satisfies the following conditions for all i,j =1,……,r. i|-.j|-=1,……s. then it is multi-symplectic and satisfies the following discrete multi-symplectic conservation lawTheorem 2 If the above RKN-RKN discrete scheme satisfies the following conditionsfor all i.j = 1,……r. i|-.j|- = 1,……,s, then it is multi-symplectic and satisfies the following discrete multi-symplectic conservation lawIn order to carry out numerical computation, we construct an explicit scheme which is equivalent to the previous discussed RKN-RKN discrete method for the linear Boussinesq equation and the beam equation, respectively. In the following two theorems we discuss their conditional stability.Theorem 3 Fixα= -1/4.β= 1/4. if there holds 0 <τ/h < 2, the equivalent explicit scheme is stable. Theorem 4 Fixα= -1/4.β= 1/4. if there holds 0 <τ/h~2 < 1. the equivalent explicit scheme is stable.Finally, we present some numerical experiments to illustrate the validity of these schemes.