The Small Dispersion Limit for a Nonlinear Semidiscrete System of Equations
The Small Dispersion Limit for a Nonlinear Semidiscrete System of Equations
复制标题
非线性半离散方程组的小色散极限
DOI：
10.1111/1467-9590.00060
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发表时间：
1997
影响因子：
2.7
通讯作者：
R. Rosales
中科院分区：
文献类型：
--
作者：
C. Turner;R. Rosales
数值实验表明，出现的振荡行为的色散方案近似的霍普夫方程的解决方案。振荡出现在同一时间，经典的解决方案的霍普夫方程开发一个奇异性，一般有一个空间周期等于两倍的网格大小。这些周期-两个解决方案的调制方程推导。调制方程具有双曲和椭圆区域。第二周期振荡在进入椭圆区域后破裂，解爆破。我们用精确解给出了爆破的局部描述。这种现象（爆破）尚未观察到的可积计划。调制方程也有不寻常的功能，他们承认（一些）冲击时，交叉的特点，在双曲区发生。其他交叉导致故障的二进制振荡的描述，与振荡行为的一个更复杂的性质出现。
Numerical experiments that illustrate the emergence of oscillatory behavior in the solutions of a dispersive scheme approximating the Hopf equation are presented. The oscillations arise at the same time that the classical solution of the Hopf equation develops a singularity and generally have a spatial period equal to twice the grid size. Modulation equations for these period‐two solutions are derived. The modulation equations have both a hyperbolic and an elliptic region. The period‐two oscillations break down after they enter the elliptic region, and the solution blows up. We give a local description of the blowup by an exact solution. This kind of phenomenon (the blowup) has not been observed for integrable schemes. The modulation equations also have the unusual feature that they admit (some) shocks when crossings of characteristics in the hyperbolic regime occur. Other crossings lead to breakdown of the binary oscillation description, with oscillatory behavior of a more complicated nature arising.