Eigenvalue Stability -- Help

 

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 equition 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 λ and the stepsize Δt .

The lower right window displays the absolute value of z as a product of Δt and | λ | , both represented in log scale.

The [θ] 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 Δt and | λ | required for stability, and how that relationship depends upon Arg(λ) .

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

A value of φ may be selected by grabbing the [φ] 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 φ has been chosen, the range of values of Δt and | λ | 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 Δt and | λ | which lead to it are displayed in orange in the lower right window.

 

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