Numerical Analysis/stability of RK methods
Definitions
[edit  edit source]Let be the local truncation error, and k is the number of time steps, then:
 The numerical method is consistent with a differential equation if
 over .
 A numerical method is said be convergent with respect to differential equation if
 over ;
 A numerical method is stable if small change in the initial conditions or data, produce a correspondingly small change in the subsequent approximations.
Theorem: For an initial value problem
 with
and certain initial conditions, , let us consider a numerical method of the form
 and .
If there exists a value such that it is continuous on the iterative domain, and if there exists an such that
 for all ,
the method fulfills the w:Lipschitz condition, and it is stable and convergent if and only if it is consistent. That is,
 for all .
For a similar argument, one can deduce the following for multi step methods:
 The method is stable if and only if all roots, , of the characteristic polynomial satisfy
 ,
 one more result is that if the method is consistent with the differential equation, the method is stable if and only if it it is convergent.
see [1]
stability polynomials of RungeKutta methods
[edit  edit source]The w:Runge–Kutta methods are very useful in solving systems of differential equations, it has wide applications for the scientists and the engineers, as well as for the economical models, the recognized with their practical accuracy where we can use and get very good results and approximations when solving an ODE problem, RK has the general form :, where
such that
 s is the number of stages,
 a_{ij} (for 1 ≤ j < i ≤ s),
 b_{i} (for i = 1, 2, ..., s)
 and c_{i} (for i = 2, 3, ..., s).
Example:finding the stability polynomial for RK4's methods
[edit  edit source]for RK4" case", which characterizes by which has the form: the stability region is found by applying the method to the linear test equation
 the stability region is found by applying the method to the linear test equation
using the linearized equation , and considering , we get;
substitute these back in , yields
 and so the characteristic polynomial
for the absolute stability region for this method, set R(z)<1, and so we get the region in figure 1. case
the table below shows the final forms for the stability function for different forms of RK4, these RK4's are different in the values of , and they are fullfilling the consistencty requirement for the method i.e :
case#  stability function 
caseI  
caseII_a  
caseII_b  
caseIII  
caseIV  
classical Rk 

see [2] to find the table
Plotting the stability region
[edit  edit source]In order to plot the stability region, we can set the stability function to be bounded by 1 and solve for the values of z, then draw z in the complex plane. Since R(z) is the unit circle in the complex plane, each point on the boundary can be represented as and so by changing over the interval, we can draw the boundaries of that region. The following w:OCTAVE/Matlab code does this by plotting contour curves until reaching the boudaries:
[x,y] = meshgrid(6:0.01:6,6:0.01:6);
z = x+i*y;
R = 1+z+0.5*z.^2+(1/6)*z.^3+(1/24)*z.^4;
zlevel = abs(R);
contour(x,y,zlevel,[1 1]);
The figure at right shows the absolute stability regions for RK4 cases which is tabulated above[3]
References
[edit  edit source] Eberly, David (2008), stability analysis for systems of defferential equation.[4]
 Ababneh, Osama; Ahmad, Rokiah; Ismail, Eddie (2009), "on cases of fourthorder RungeKutta methods", European journal of scientific Research.[5]
 Mathews, John; Fink, Kurtis (1992), numerical methods using matlab.
 Stability of RungeKutta Methods [6]