Chebyshev谱配置法的稳定性与收敛性基础(1)

数值格式被称为是稳定的如果数值解可以被不依赖于\(N\)(多项式的次数)的数据所控制。也就是说数值解的范数可以被某一个上界所控制，这个上界是一个与\(N\)无关的常数与已知数据(初值或右端)的乘积。

稳定性通过能量方法或者广义变分原理来证明。

考虑问题
\[\frac{\partial u}{\partial t}=\frac{\partial u}{\partial x^2},\quad -1<x<1,t>0\]
边界条件
\[u(-1,t)=u(1,t)=0\]
初始条件
\[u(x,0)=u_0(x)\]
使用配置法求解时，有如下的配置方程
\[\frac{\partial u^N}{\partial t}(x_k,t)=\frac{\partial u}{\partial x^2}(x_k,t),\quad k=1,\dots,N-1\tag{1}\]
其中\(x_k\)是配置点。 使用能量方法，将上式两边都乘以\(u^N(x_k,t)\),并求和，得到

\[\sum_{k=0}^{N}\frac{\partial u^N}{\partial t}(x_k,t)u^N(x_k,t)\omega_k=\sum_{k=0}^{N}\frac{\partial u^N}{\partial x^2}(x_k,t)u^N(x_k,t)\omega_k\]
进一步可以写为
\[\frac12\frac{d}{dt}\sum_{k=0}^N [u^N(x_k,t)]^2\omega_k=\sum_{k=0}^{N}\frac{\partial u^N}{\partial x^2}(x_k,t)u^N(x_k,t)\omega_k\tag{2}\]
因为\(u^N\)是\(N\)次多项式，\(\frac{\partial^2 u}{\partial x^2}u^N\)是\(2N-2\)的多项式。对于高斯型求积公式，有

定理 对于插值型求积公式
\[\int_{-1}^1\rho(x)f(x)dx\approx\sum_{i=1}^N A_i f(x_i) \]
具有\(2N-1\)次代数精度，当且仅当插值节点\(x_1,x_2,\dots.x_N\)是\([a,b]\)上以\(\rho(x)\)为权的正交多项式的零点。

因此，式(2)中的右端可以精确的写成
\[\int_{-1}^1 \frac{\partial u^N}{\partial x^2}(x_k,t)u^N(x_k,t)\omega_k dx=\sum_{k=0}^{N}\frac{\partial u^N}{\partial x^2}(x_k,t)u^N(x_k,t)\omega_k\tag{3}\]
结合(2)(3)式，有
\[\frac12\frac{d}{dt}\sum_{k=0}^N [u^N(x_k,t)]^2\omega_k= \int_{-1}^1 \frac{\partial u^N}{\partial x^2}(x_k,t)u^N(x_k,t)\omega_k dx\tag{4}\]

先上一个结论(引理)
\[-\int_{-1}^{1}\frac{\partial^2 u^N}{\partial x^2}(x,t)u^N(x,t)\omega(x)dx\geq \frac14\int_{-1}^{1}[\frac{\partial u^N}{\partial x}(x,t)]^2\omega(x)dx\tag{5}\]
结合(4)和(5)
\[\frac12\frac{d}{dt}\sum_{k=0}^N [u^N(x_k,t)]^2\omega_k+\frac14\int_{-1}^{1}[\frac{\partial u^N}{\partial x}(x,t)]^2\omega(x)dx\leq 0\]

在\([0,t]\)上对上式两边积分
\[\frac12 \sum_{k=0}^N [u^N(x_k,t)]^2\omega_k+\frac14\int_0^t\int_{-1}^{1}[\frac{\partial u^N}{\partial x}(x,t)]^2\omega(x)dxds-\sum_{k=0}^N[u_0(x_k,t)]^2 \leq 0\]
亦
\[\frac12 \sum_{k=0}^N [u^N(x_k,t)]^2\omega_k+\frac14\int_0^t\int_{-1}^{1}[\frac{\partial u^N}{\partial x}(x,t)]^2\omega(x)dxds \leq \sum_{k=0}^N[u_0(x_k,t)]^2\]

所以

\[\frac12 \sum_{k=0}^N [u^N(x_k,t)]^2\omega_k+\frac14\int_0^t\int_{-1}^{1}[\frac{\partial u^N}{\partial x}(x,t)]^2\omega(x)dxds \leq 2 \max\limits_{-1\leq x\leq 1}|u_0(x_k,t)|^2\]

这就证明了Chebyshev配置法(关于初值)的稳定性。

参考文献

spectral and high-order methods with application