EIGENVALUE STABILITY

This tool illustrates the concepts surrounding the eigenvalue analysis of the stability of various difference schemes.

Select a difference scheme using the dropdown menu at lower left. The equation relating the "gain" $$g$$ of each step in the numerical iteration to the "discrete eigenvalue" $$z$$ is revealed using the [Formula] key. Options: Forward Euler (RK1), Midpoint, Backwards Euler, Trapezoidal, RK2, RK4.

The upper right window displays in cyan the unit circle in the $$g$$ plane, and the unit disk in dark blue.

The left hand window displays in dark blue the stable range of values of $$z$$: $$z$$ such that all corresponding $$g$$'s have absolute value less than $$1$$. The $$z$$'s with absolute value equal to $$1$$ are shown in cyan.

$$z$$ is the product of the eigenvalue $$\lambda$$ and the stepsize $$\Delta t$$.

The lower right window displays the absolute value of $$z$$ as a product of $$\Delta t$$ and $$|\lambda|$$, both represented in $$\log$$ scale.

The $$[\theta]$$ slider to the right of the $$g$$ plane controls the argument of a point on the unit circle. It and the corresponding values of $$z$$ are plotted in yellow.

The relationship between $$z$$ and $$g$$ can be further explored by dragging the cursor over the $$g$$ plane or over the $$z$$ plane.

The applet demonstrates the relationship between $$\Delta t$$ and $$|\lambda|$$ required for stability, and how that relationship depends upon $$\text{Arg}(\lambda)$$.

The values of $$\Delta t$$ and $$|\lambda|$$ can be adjusted using sliders adjacent to the lower right window or by dragging the cursor inside that window. If the values of $$\lambda$$; and $$\Delta t$$ have been chosen, the product $$|\lambda| \Delta t$$ is recorded under this window. If no value of $$\phi$$; has been selected, the corresponding circle of values of $$z$$ is shown in the upper left window.

A value of $$\phi$$ may be selected by grabbing the $$[\phi]$$ slider. A green ray at the chosen polar angle is shown in the $$z$$ plane. If a value of $$|z|$$has been selected, the corresonding value of $$z$$ is marked with an orange point in the $$g$$ plane, and the corresponding values of $$g$$ are marked in the $$g$$ plane and connected to the origin by struts.

If a value of $$\phi$$ has been chosen, the range of values of $$\Delta t$$ and $$|\lambda|$$ leading to stable $$z$$ are shown in dark blue in the lower right window, and the values leading to $$|g| = 1$$ are shown in cyan.

If a value of $$|z|$$ has also been chosen, the locus of values of $$\Delta t$$ and $$|\lambda|$$ which lead to it are displayed in orange in the lower right window.

© 2010 C. Lieberman, H. Miller, J.-M. Claus