数学
时间尺度微积分
非线性系统
有限差分
不变(物理)
微分方程
有限差分法
守恒定律
应用数学
数学分析
牙石(牙科)
数学物理
医学
物理
牙科
量子力学
多元微积分
控制工程
工程类
摘要
In a previous work, the authors have presented a formalism for deriving systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities. This formalism is herein cast in the context of exact finite difference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus, in the same sense as that of the corresponding differential equation being derivable from its associated energy function (a conserved quantity). A clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities, which are the consistent counterpart of the invariant quantities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the context of finite difference calculus. Some numerical experiments are presented to illustrate the conservation property of the proposed algorithms.
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