MarkFL said:Here is an article explaining how to plot this stability region:
http://homepages.cwi.nl/~jason/Classes/numwisk/ch10.pdf
MarkFL said:Beautiful image, Dustin! (Clapping)
Did you write your own contourplot function?
[X, Y] = meshgrid(-3:.01:1, -3:.01:3);
Mu = X + i*Y;
R = 1 + Mu + .5*Mu.^2 + (1/6)*Mu.^3 + (1/24)*Mu.^4;
Rhat = abs(R);
Rhat = Rhat.*(Rhat<1); %# here I truncate
imagesc([min(X(:)) max(X(:))],[min(Y(:)) max(Y(:))], Rhat)
colormap(flipud(bone))The purpose of plotting the Runge Kutta 4 stability region is to analyze the stability of the numerical integration method. By visualizing the stability region, we can determine the range of step sizes and values of the differential equation parameters for which the method will produce accurate results.
The shape of the stability region indicates the region in which the method is stable and will produce accurate results. A larger stability region indicates that the method is more accurate and can handle a wider range of parameter values and step sizes.
The stability regions of different numerical integration methods can vary depending on the order and type of the method. Generally, higher order methods have larger stability regions and can handle more complex and stiff differential equations. However, the stability region of a specific method may also be affected by the step size and parameters of the differential equation being solved.
Choosing a stable step size is important because it ensures that the numerical solution will be accurate and not diverge or oscillate. If the step size is too large, the method may become unstable and produce incorrect results. Therefore, selecting a step size within the stability region is crucial for the accuracy of the numerical solution.
Plotting the stability region for the Runge Kutta 4 method is limited to two dimensions, which may not fully represent the stability of the method for all values of the parameters and step sizes. Additionally, the stability region may vary for different types of differential equations and may not accurately reflect the stability for a specific problem being solved.