摘要
In this paper, we introduce the multisymplecticstructure of the nonlinear wave equation, and prove that theclassical five-point scheme for the equation is multisymplec-tic. Numerical simulations of this multisymplectic scheme onhighly oscillatory waves of the nonlinear Klein-Gordonequation and the collisions between kink and anti-kink soli-tons of the sine-Gordon equation are also provided. The mul-tisymplectic schemes do not need to discrete PDEs in thespace first as the symplectic schemes do and preserve notonly the geometric structure of the PDEs accurately, but alsotheir first integrals approximately such as the energy, themomentum and so on. Thus the multisymplectic schemeshave better numerical stability and long-time numerical be-havior than the energy-conserving scheme and the symplec-tic scheme.
In this paper, we introduce the multisymplectic structure of the nonlinear wave equation, and prove that the classical five-point scheme for the equation is multisymplec- tic. Numerical simulations of this multisymplectic scheme on highly oscillatory waves of the nonlinear Klein-Gordon equation and the collisions between kink and anti-kink soli- tons of the sine-Gordon equation are also provided. The mul- tisymplectic schemes do not need to discrete PDEs in the space first as the symplectic schemes do and preserve not only the geometric structure of the PDEs accurately, but also their first integrals approximately such as the energy, the momentum and so on. Thus the multisymplectic schemes have better numerical stability and long-time numerical be- havior than the energy-conserving scheme and the symplec- tic scheme.
