论文标题
缀合物符号方法的向后错误分析
Backward error analysis for conjugate symplectic methods
论文作者
论文摘要
普通微分方程的数值解可以解释为附近修改方程的精确解。通过分析修改方程来研究数值解的行为称为向后误差分析。如果原始方程共享结构属性，则精确和近似的解决方案具有几何特征，例如保守量的存在。当应用于哈密顿系统时，共轭符号方法可以保留修改的符号形式和修饰的哈密顿量。我们展示了如何使用变分和符号技术的混合版本来计算修改的符号和汉密尔顿结构。与其他方法相反，我们的后退误差分析方法不依赖ANSATZ，而是系统地计算结构，前提是该方法的变异表述是已知的。该技术在具有矩阵系数的对称线性多步法方法的示例中进行了说明。
The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.
