# what is the significance of $\kappa$ in the proof of inconditional stability of the Crank-Nicolson scheme

When using a centered difference approximation

$$\frac{\partial}{\partial t}u(t,x) = \frac{u(t + \Delta t/2,x) - u(t - \Delta t/2, x)}{\Delta t} + O((\Delta t)^2)$$

It is an approximation of the differential equation $$\dot{u} = ku''$$ at the midpoint $$(\frac{t_i + t_{i+1}}{2}, x_j)$$. The scheme is consistent with the error $$O((\Delta t)^2) + O((\Delta x)^2)$$.

$$\frac{u_{i+1, j}-u_{i,j}}{\kappa \Delta t} = \frac{u_{i,j-1} - 2u_{i,j} + u_{i,j + 1}}{2(\Delta x)^2} + \frac{u_{i+1,j-1} - 2u_{i+1,j} + u_{i+1,j + 1}}{2(\Delta x)^2}$$

$$\vec{u}_{i+1} - \vec{u}_i = -\frac{\kappa \Delta t}{2}(\textbf{A}_n \cdot \vec{u}_{i+1} + \textbf{A}_n \cdot \vec{u}_i)$$

or

$$\Big(\mathbb{I}_n + \frac{\kappa \Delta t}{2}\textbf{A}_n\Big) \cdot \vec{u}_{i+1} = \Big(\mathbb{I}_n - \frac{\kappa \Delta t}{2}\textbf{A}_n\Big) \cdot \vec{u}_i$$

If the values $$\vec{u}_i$$ at a given time are known we have to multiply the vector with a matrix and then solve a system of linear equations to determine the values $$\vec{u}_{i+1}$$ at the next time level. We have an implicit method. We can use the eigenvalues and vectors of $$\textbf{A}_n$$ to examine stability of the scheme. We are lead to examine the inequality

$$\Bigg|\frac{2 - \kappa \Delta t \lambda_k}{2 + \kappa \Delta t \lambda_k}\Bigg|^2 < 1$$

Since $$\lambda_k > 0$$ we find that this scheme is $$\textbf{unconditionally stable.}$$

My question

what is the significance of the $$\kappa$$ in that proof? is it a condition number of a matrix? if yes which one?

• It is most probably that $\kappa = k$ in the PDE $$\dot{u} = \color{red}{\kappa}u''$$ and that the $1/(\Delta x)^{2}$ term has been absorbed into the A matrix. – Mattos Dec 3 '18 at 1:41