Ngô Quốc Anh

August 16, 2009

On the stability of the Runge-Kutta 4 (RK4)

In the literature, the so-called RK4 is given as following: we first define the following coefficients

\displaystyle \begin{gathered}{K_{1}}= f\left({{t_{i}},{y_{i}}}\right),\hfill\\ {K_{2}}= f\left({{t_{i}}+\frac{1}{2}\Delta t,{y_{i}}+\frac{1}{2}\Delta t{K_{1}}}\right),\hfill\\ {K_{3}}= f\left({{t_{i}}+\frac{1}{2}\Delta t,{y_{i}}+\frac{1}{2}\Delta t{K_{2}}}\right),\hfill\\ {K_{4}}= f\left({{t_{i}}+\Delta t,{y_{i}}+\Delta t{K_{3}}}\right),\hfill\\ \end{gathered}

then

\displaystyle {y_{i + 1}} = {y_i} + \frac{{\Delta t}} {6}\left( {{K_1} + 2{K_2} + 2{K_3} + {K_4}} \right).

This is the most important iterative method for the approximation of solutions of ordinary differential equations

\displaystyle y' = f(t, y), \quad y(t_0) = y_0.

This technique was developed around 1900 by the German mathematicians C. Runge and M.W. Kutta. In order to study its stability, we use the model problem

\displaystyle y' = \lambda y, \quad \Re \lambda < 0.

In other words, we replace f(t,y) by \lambda y. Then the stability condition for time step \Delta comes from the following condition

\displaystyle \left| \frac{y_{i+1}}{y_i} \right| \leq 1.

Applying the above discussion to RK4 method, we see that

\displaystyle\begin{gathered}{K_{4}}=\lambda\left({{y_{i}}+\Delta t{K_{3}}}\right),\hfill\\ {K_{3}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{K_{2}}}\right),\hfill\\ {K_{2}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{K_{1}}}\right),\hfill\\ {K_{1}}=\lambda{y_{i}},\hfill\\ \end{gathered}

which implies

\displaystyle\begin{gathered}{K_{4}}=\lambda\left({{y_{i}}+\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{y_{i}}}\right)}\right)}\right) =\lambda{y_{i}}\left({1+\Delta t\lambda\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right)}\right),\hfill\\ {K_{3}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{y_{i}}}\right)}\right) =\lambda{y_{i}}\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right),\hfill\\ {K_{2}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda{y_{i}}}\right) =\lambda{y_{i}}\left({1+\frac{1}{2}\Delta t\lambda }\right),\hfill\\ {K_{1}}=\lambda{y_{i}},\hfill\\ \end{gathered}

which yields

\displaystyle\begin{gathered}{K_{4}}=\lambda\left({{y_{i}}+\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{y_{i}}}\right)}\right)}\right) =\lambda{y_{i}}\left({1+\Delta t\lambda\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right)}\right),\hfill\\ {K_{3}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{y_{i}}}\right)}\right) =\lambda{y_{i}}\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right),\hfill\\ {K_{2}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda{y_{i}}}\right) =\lambda{y_{i}}\left({1+\frac{1}{2}\Delta t\lambda }\right),\hfill\\ {K_{1}}=\lambda{y_{i}},\hfill\\ \end{gathered}

Thus

\displaystyle\frac{{{y_{i+1}}}}{{{y_{i}}}}= 1+\frac{{\Delta t}}{6}\left[{\lambda+2\lambda\left({1+\frac{1}{2}\Delta t\lambda }\right)+2\lambda\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right)+\lambda\left[{1+\Delta t\lambda\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right)}\right]}\right]

which is nothing but

\displaystyle\frac{{{y_{i+1}}}}{{{y_{i}}}}= 1+\Delta t\lambda+\frac{{{{\left({\Delta t\lambda }\right)}^{2}}}}{2}+\frac{{{{\left({\Delta t\lambda }\right)}^{3}}}}{6}+\frac{{{{\left({\Delta t\lambda }\right)}^{4}}}}{{24}}.

Therefore, the stability condition is given as follows

\displaystyle \left| {1 + z + \frac{{{z^2}}} {2} + \frac{{{z^3}}} {6} + \frac{{{z^4}}} {{24}}} \right| \leq 1, \quad \Re z < 0.

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