论文标题
偏微分方程的变异离散性的向后错误分析
Backward error analysis for variational discretisations of partial differential equations
论文作者
论文摘要
在向后误差分析中，将方程式的近似解与附近修改方程的精确解决方案进行了比较。在数值的普通微分方程中，两者同意步长的任何功能。如果微分方程具有几何属性，则修改后的方程可能会共享。这样，可以将微分方程的已知属性应用于近似值。但是对于部分微分方程，已知的修改方程是高阶，限制了理论的适用性。因此，我们研究了由离散的变分原理产生的离散偏微分方程的对称解。这些对称溶液遵守无限二维功能方程。我们表明，这些方程式允许二阶修改方程式，这些方程式是哈密顿量，也具有修改后的坐标中的一阶拉格朗日。针对非线性波方程中旋转行驶波的情况，对修改的方程式及其相关结构进行了明确计算。
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.
