In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby modified equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.
翻译:在后向错误分析中,对等方程式的大致解决办法与附近修改的方程式的精确解决办法比较。在数字普通差分方程式中,两个方程式同意步数大小的任何功率。如果差分方程式具有几何属性,则修改后方程式可以分享。这样,差异方程式的已知特性可以适用于近似值。但是,对于部分差分方程式,已知的修改后方程式的等级较高,限制了理论的适用性。因此,我们研究离散部分差方程式的对称办法,这些对称办法来自离散的变异原则。这些对称办法符合无限维功能方程式。我们显示,这些方程式接受二级修改的方程式,它们是汉密尔顿式,在修改后的坐标中也拥有一级拉格朗方程式。在非线性波方程式中旋转流动波的情况下,对修改后的方程式及其相关结构进行了明确的计算。