Example 4 Given an integer $n \geqslant 2$. Find the largest constant $\lambda(n)$ such that if the real sequence $a_{0}, a_{1}, \cdots, a_{n}$ satisfies:
$$
\begin{array}{l}
0=a_{0} \leqslant a_{1} \leqslant \cdots \leqslant a_{n}, \\
2 a_{i} \geqslant a_{i+1}+a_{i-1}, \\
\text { then }\left(\sum_{i=1}^{n} i a_{i}\right)^{2} \geqslant \lambda(n) \sum_{i=1}^{n} a_{i}^{2} .
\end{array}
$$