# stable solutions for Large-scale ODEs under boundary value problem

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!

• I'm not familiar with the stability conditions of DQM, but have you checked the eigenvalues of A? If any of them have a positive real component than that means some mode is growing exponentially. You also need to make sure that all the eigenvalues of A fit into the region of stability for whatever ODE solver you are using. Jul 14, 2022 at 14:56