Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs

Table 1

The error and convergence orders of (3.9) for Klein-Gordon problem with Δ𝑡=Δ𝑥.

Mesh size	𝑡=1	𝑡=1.5	𝑡=2
Δ𝑥=0.1
Error	6.3𝑒−3	8.9𝑒−3	1.07𝑒−2
Order	—	—	—
Δ𝑥=0.05
Error	1.6𝑒−3	2.3𝑒−3	2.8𝑒−3
Order	1.9773	1.9522	1.9341
Δ𝑥=0.025
Error	4.1645𝑒−4	5.8258𝑒−4	7.0059𝑒−4
Order	1.9419	1.9811	1.9988
Δ𝑥=0.0125
Error	1.0480𝑒−4	1.4622𝑒−4	1.7565𝑒−4
Order	1.9905	1.9943	1.9959