三阶Airy方程满足两个守恒律的非线性差分格式

Nonlinear difference scheme which conserves the first two conservation laws for third-order Airy equation

近年来,学者们对发展型偏微分方程设计了一种能保持多个守恒律的数值方法,这类方法无论在解的精度还是长时间的数值模拟方面都表现出非常好的性质.将这类思想应用到三阶Airy方程,即三阶散射方程,对其设计了满足两个守恒律的非线性差分格式.该格式不仅计算数值解,同时计算数值能量,并且保证数值解和数值能量同时守恒.从数值结果可以看出,该格式在长时间的数值模拟中具有更好的保结构性质.

In recent years,researchers developed a new type of numerical method which can satisfy multi-conservation laws for the evolutionary partial differential equations.The new methods showed an extremely outstanding quality in both accuracies and long-time behaviors of numerical solutions.We apply these ideas into the third-order Airy equation,developing a nonlinear difference scheme which can satisfy two conservation laws.Our scheme will compute not only numerical solutions but also numerical energy and maintain the two conservation laws of them.The numerical results show that the scheme has a good quality in long-time numerical simulations.