谱方法

Matlab中的谱方法Outline谱方法的应用领域有限差分应用领域常微分方程和偏微分方程主要包括：流体力学，量子力学，振动力学，线性和非线性波，复杂分析等其它领域。有限差分近似考虑一些均匀的网格其中与之对应的值为},,{1Nxxhxxjj1},,{1Nuu1uNu2u2x1xNx1u2u1u1x2u1u2x1x2u1uNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1uNuNx2x1x2u1u1u2u1u1u1x1x1u1x2u1u1x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1xNx2x2u1u1xNuNx2x2u1u1xNuNx2x2u1u1x1x1u1x2u1u1x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1x2x2u1u1xNx2x2u1u1xNuNx2x2u1u1xNuNx2x2u1u1x利用二阶有限差分近似方法：考虑具有周期的格点：描述这个过程可以用矩阵表示：)('jjxuwhuuwjjj211Nuu011Nuuhuuwjjj211Nuu0huuwjjj21111NuuNuu0huuwjjj21111NuuNuu0huuwjjj211推导过程对于是度小于2的多项式，满足设置对于固定点内插值为：其中Nj,,2,1jp1111)(,)(,)(jjjjjjjjjuxpuxpuxp)('jjjxpwjjp)()()()(11011xauxauxauxpjjjj2211011211()()()/2,()()()/()()()/2jjjjjjaxxxxxhaxxxxxhaxxxxxh)()()()(11011xauxauxauxpjjjj1111)(,)(,)(jjjjjjjjjuxpuxpuxp1111)(,)(,)(jjjjjjjjjuxpuxpuxpNj,,2,12211011211()()()/2,()()()/()()()/2jjjjjjaxxxxxhaxxxxxhaxxxxxh产生更高阶的方法对于是度小于4的多项式，满足设置任然假设数据具有周期性，则可以得到矢量矩阵如下图所示Nj,,2,1jp2211(),(),()jjjjjjjjjpxupxupxu)('jjjxpw例子)sin()(xexu],[在内产生周期序列：1x2xNx1x2x1x2x1x2x1x2x1x2x1xNx2x1x2x1xNx2x1xNx2x1xNx2x1x程序（Matlab)图形对于无限的等间距网格对于有限网格让p是与j无关的单值函数，比如设置jjuxp)()('jjxpw谱配置方法设计规则对于周期区域在一个等间距的网格中采用三角多项式，非周期区域采用代数多项式。以周期稠密矩阵（N*N）为例N是偶数图形summary程序一和程序二大致上一样，只是矩阵发生了变化。图二中误差减小的非常快，这种行为称为谱方法的准确性。已知在网格中的离散数据，通过全局插值，然后估计在网格中插值的导数，对于周期问题，我们一般在等间距格点位置上使用三角函数多项式插值，相反在非周期非等间距的格点处使用多项式插值。附录多项式插值三角插值THEEND谢谢！