An artificial boundary condition for the multisymplectic Preissman scheme

Recently the Preissman scheme has become very popular [Zao and Qin: J. Phys. A 33 (2000) 3613; Chen: Appl. Math. Comput. 124 (2001) 371; Reich: Reading Material of Int. Workshop on Structure-Preserving Algorithms, 2001, Vol. 6, No. 3, pp. 153-174; Wang and Qin: Math. Comput. Model. 36 (2002) 963] since it was found to be a multisymplectic scheme [Bridges: Math. Proc. Cambridge Philos. Soc. 121 (1997) 147]. However, When it is applied to solve the periodic boundary problem of the soliton equations with degenerate lagrangian like the Korteweg-De Vries equation, the Kadomtsev-Petviashvili equation and the water waves equations, the general widely-used iterative method refereed as the simple iterative method [Wineberg et al.: J. Comput. Phys. 97 (1991) 311] is not convergent, so are the other iterative method such as Newton method and conjugate gradient method. Why? This problem has puzzled researchers all along. In this paper, taking the KdV equation as an example, we analyze this problem and show that the unconvergence is due to indeterminacy of the introduced potential function. An artificial numerical condition is added to the periodic boundary condition. The added boundary condition makes the numerical implementation of the multisymplectic Preissman scheme practical and is proved not to change the primary numerical solutions of the KdV equation. Numerical experiments on soliton collisions, with comparisons of the spectral method and the Zabusky-Kruskal scheme, are also provided to confirm our conclusion and show the merits of the multisymplectic scheme.