We consider the periodic initial-value problem for the Korteweg-de Vries equation that we discretize in space by a spectral Fourier-Galerkin method and in time by an implicit, high order, Runge-Kutta scheme of composition type based on the implicit midpoint rule. We prove $L^{2}$ error estimates for the resulting semidiscrete and the fully discrete approximations.

翻译：我们考虑了Korteweg-de Vries等式的定期初始价值问题,我们用光谱Fourier-Galerkin方法在空间分解,并及时根据隐含中点规则,以隐含的、高顺序、龙格-库塔构成类型计划的方式分解。 我们证明对由此产生的半分异物和完全离散近似物的误差估计值为$L%2}。