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)$.

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

The matrix notation leads to

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


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

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?

  • 1
    $\begingroup$ 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. $\endgroup$ – Mattos Dec 3 '18 at 1:41

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