分区平均向量场方法
Partitioned AVF methods
AI总结：
本文提出一种分区一阶AVF方法，通过变量分组逐步应用AVF方法，结合伴随方法提升精度至二阶，并构造经典模型方程的显式方案，具有更低计算成本和额外守恒性质。
摘要：
经典的二阶平均向量场（AVF）方法能精确保持哈密顿常微分方程和偏微分方程的能量。然而，AVF方法不可避免地导致一般非线性系统中全隐式非线性代数方程。为克服这一缺点并保持所需的能量保持性质，提出了一种一阶分区AVF方法，首先将变量分为若干组，然后逐步应用AVF方法。结合其伴随方法，分别提出分区AVF组合方法和加法方法以提升其精度至二阶。针对两个经典模型方程构造了具体的方案，具有半隐式、线性隐式性质，其计算成本显著低于原始AVF方法。此外，对于特定问题，除了常规的能量保持外，还可生成额外的守恒性质。这些方案的数值验证进一步证实了我们的结果。
英文摘要：
The classic second-order average vector field (AVF) method can exactly preserve the energy for Hamiltonian ordinary differential equations and partial differential equations. However, the AVF method inevitably leads to fully-implicit nonlinear algebraic equations for general nonlinear systems. To address this drawback and maintain the desired energy-preserving property, a first-order partitioned AVF method is proposed which first divides the variables into groups and then applies the AVF method step by step. In conjunction with its adjoint method we present the partitioned AVF composition method and plus method respectively to improve its accuracy to second order. Concrete schemes for two classic model equations are constructed with semi-implicit, linear-implicit properties that make considerable lower cost than the original AVF method. Furthermore, additional conservative property can be generated besides the conventional energy preservation for specific problems. Numerical verification of these schemes further conforms our results.
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