Runge-Kutta

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node9.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node7.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node11.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node10.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H2>Runge-Kutta</H2> <P> (Pardon this sounding like a television commercial, it just happens to be true.) During the instructor’s graduate school days the standard differential equation solver was the fourth-order Runge-Kutta method. It has more precision than Euler’s algorithm, is easy to start and program, and is relatively fast. This classic then lost its appeal in favor of more intelligent and self-correcting methods. Yet with experience, the faults in these other methods have left them less attractive so that many computational scientists are again returning to Runge-Kutta. In fact, a fourth-order Runge-Kutta with a self-adjusting step size has proven itself to be robust and capable of industrial-strength work. In this section we derive the second-order Runge-Kutta method and give the results and a sample subroutine for the fourth-order method. <P> A approach to developing an algorithm for integrating a differential equation is to write down the formal integral of the equation and then approximate the integral: <P> <IMG WIDTH=500 HEIGHT=86 ALIGN=BOTTOM ALT="[formal soln.]" SRC="img70.gif"> <P> The Runge-Kutta method approximates the formal solution (64) by expanding <nobr><i>f</i>(<i>t</i>,<i>y</i>)</nobr> in a Taylor series about the <EM>midpoint</EM> of the integration subinterval, <P> <IMG WIDTH=500 HEIGHT=35 ALIGN=BOTTOM ALT="[RK approx.]" SRC="img71.gif"> <P> Yet since the integral of <IMG WIDTH=80 HEIGHT=18 ALIGN=ABSMIDDLE ALT="t - t_(n + 1/2)" SRC="img72.gif"> vanishes when evaluated about the midpoint, we automatically get improved precision using only the first term in (65): <P> <IMG WIDTH=500 HEIGHT=68 ALIGN=BOTTOM ALT="[RK2 derivation]" SRC="img73.gif"> <P> This algorithm effectively uses the derivative of the function evaluated at the midpoint to predict the solution at the end of the subinterval (Note that the algorithm does not use the midpoint again when computing the next interval since we want the freedom to employ adaptive step size, that is, <i>h</i> may not be the same for the next interval). Since the error in (68) is <IMG WIDTH=40 HEIGHT=16 ALIGN=TOP ALT="O(h^3)" SRC="img74.gif"> it is more accurate than Euler’s algorithm. Yet (65) cannot be applied immediately since it requires a knowledge of <IMG WIDTH=45 HEIGHT=15 ALIGN=MIDDLE ALT="y_(n + 1/2)" SRC="img75.gif"> which is not in the scheme of things. We thus approximate <IMG WIDTH=45 HEIGHT=15 ALIGN=ABSMIDDLE ALT="y_(n + 1/2)" SRC="img75.gif"> with Euler’s algorithm (which may only be <IMG WIDTH=40 HEIGHT=16 ALIGN=TOP ALT="O(h^2)" SRC="img76.gif">, but when multiplied by the <i>h</i> in (65), keeps our error <IMG WIDTH=40 HEIGHT=16 ALIGN=TOP ALT="O(h^3)" SRC="img74.gif"> ): <P> <IMG WIDTH=500 HEIGHT=129 ALIGN=BOTTOM ALT="[deriv. of RK2]" SRC="img78.gif"> <P> The second-order Runge-Kutta algorithm (71) requires the known derivative function <i>f</i> at the endpoints and midpoint of the interval, and the unknown function <i>y</i> at the previous point. Since we start with initial conditions, the algorithm is self starting. Note too that it is applicable with a general function <i>f</i> (for example nonlinear), and simple to program.  (If the simultaneous equations were actually higher-order differential equations, we would simultaneously be solving for some intermediate values of the derivative functions, and so the intermediate values of some <IMG WIDTH=23 HEIGHT=30 ALIGN=MIDDLE ALT="y^(i)'s" SRC="img79.gif">’s might need be approximated.) <P> As an <b>example</b>, we apply this algorithm to our spring problem for the first interval: <P> <IMG WIDTH=500 HEIGHT=388 ALIGN=BOTTOM ALT="[appl. of RK2]" SRC="img80.gif"> <P> This looks more complicated than it would be to program.  We continue using these same equations to progressively step (integrate) outwards. Note that, at last, the algorithm’s equation for the distance covered in one step is as sophisticated as the one known by all first-year physics students. <P> As indicated earlier, the <b>fourth-order Runge-Kutta method</b> is generally considered to provide an excellent balance of power, precision <nobr>( <IMG WIDTH=40 HEIGHT=16 ALIGN=TOP ALT="O(h^5)" SRC="img81.gif"> ),</nobr> and simplicity to program. Here there are four gradient or “<i>k</i>” terms which provide a better approximation to the behavior of <nobr><i>f</i>(<i>t</i>,<i>y</i>)</nobr> near the midpoint <nobr><IMG WIDTH=109 HEIGHT=18 ALIGN=ABSMIDDLE ALT="(t_(n + 1/2), y_(n + 1/2)" SRC="img82.gif"> ;</nobr> the extra gradients can still be programmed simply with just four calls to the same subroutine: <P> <A NAME="rk4"><IMG WIDTH=524 HEIGHT=115 ALIGN=BOTTOM ALT="[RK4]" SRC="img83.gif"></A> <P> <HR><center> <table border=1> <td><A HREF="node9.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node7.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node11.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node10.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> </BODY>