I'm doing FEM and have a problem about getting numerically stable solution for ODEs problems like: $$ \frac{\mathrm{d}}{\mathrm{d}x}\mathbf{Y} = \mathbf{AY}, x\in[x_1,x_2]$$ in which $\mathbf{Y}$ stands for a large-scale vector (greater than $1000000\times1$), and $\mathbf{A}$ is its corresponding coefficient matrix. The elements in matrix $\mathbf{A}$ vary greatly, from $10^{-50}$ to $10^{50}$.
For small matrix, analytical solutions for such ODEs are simple. However, those does not work for large scale matrix, especially when the transfer distance $(x_2-x_1)$ is large. I used DQM to get a relatively stable solution, but still is not stable enough when the dimension of $\mathbf{A}$ gets bigger.
Is there any stable solutions/ideas for this problem? Thanks!
