Multi-symplectic methods for membrane free vibration equation

In this paper, the multi-symplectic formulations of the membrane free vibration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-seven-points scheme with certain discrete conservation laws—a multi-symplectic conservation law (CLS), a local energy conservation law (ECL) as well as a local momentum conservation law (MCL)—is constructed to discrete the PDEs that are derived from the membrane free vibration equation. The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior. Project supported by the National Natural Science Foundation of China (Nos. 10632030 and 10572119), Program for New Century Excellent Talents of Ministry of Education of China (No. NCET-04-0958) and the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment