# Runge-Kutta 4(5)

*Runge-Kutta* is a numerical solver providing an efficient explicit method to solve Ordinary Differential Equations (ODEs) Initial Value Problems.

### Description

Called by xcos, *Runge-Kutta* is a numerical solver providing an efficient fixed-size step method to solve Initial Value Problems of the form:

*CVode* and *IDA* use variable-size steps for the integration.

A drawback of that is the unpredictable computation time. With *Runge-Kutta*, we do not adapt to the complexity of the problem, but we guarantee a stable computation time.

As of now, this method is explicit, so it is not concerned with Newton or Functional iterations, and not advised for stiff problems.

It is an enhancement of the Euler method, which approximates
*y _{n+1}*
by truncating the Taylor expansion.

By convention, to use fixed-size steps, the program first computes a fitting *h* that approaches the simulation parameter max step size.

An important difference of *Runge-Kutta* with the previous methods is that it computes up to the fourth derivative of *y*, while the others only use linear combinations of *y* and *y'*.

Here, the next value is determined by the present value
*y _{n}*
plus the weighted average of four increments, where each increment is the product of the size of the interval,

*h*, and an estimated slope specified by the function

*f(t,y)*:

*k1*is the increment based on the slope at the beginning of the interval, using*y*(Euler's method),_{n}*k2*is the increment based on the slope at the midpoint of the interval, using*y*,_{n}+ h*k1/2*k3*is again the increment based on the slope at the midpoint, but now using*y*_{n}+ h*k2/2*k4*is the increment based on the slope at the end of the interval, using*y*_{n}+ h*k3

We can see that with the *ki*, we progress in the derivatives of
*y _{n}*
. So in

*k4*, we are approximating

*y*, thus making an error in

^{(4)}_{n}*O(h*.

^{5})So the total error is
*number of steps * O(h ^{5})*
. And since

*number of steps = interval size / h*by definition, the total error is in

*O(h*.

^{4})That error analysis baptized the method *Runge-Kutta 4(5)*,
*O(h ^{5})*
per step,

*O(h*in total.

^{4})Although the solver works fine for max step size up to
*10 ^{-3}*
, rounding errors sometimes come into play as we approach

*4*10*. Indeed, the interval splitting cannot be done properly and we get capricious results.

^{-4}### Examples

The integral block returns its continuous state, we can evaluate it with *Runge-Kutta* by running the example:

// Import the diagram and set the ending time loadScicos(); loadXcosLibs(); importXcosDiagram("SCI/modules/xcos/examples/solvers/ODE_Example.zcos"); scs_m.props.tf = 5000; // Select the solver Runge-Kutta and set the precision scs_m.props.tol(6) = 6; scs_m.props.tol(7) = 10^-2; // Start the timer, launch the simulation and display time tic(); try xcos_simulate(scs_m, 4); catch disp(lasterror()); end t = toc(); disp("Time for Runge-Kutta:", t);

The Scilab console displays:

Time for Runge-Kutta: 8.906

Now, in the following script, we compare the time difference between *Runge-Kutta* and *CVode* by running the example with the five solvers in turn:
Open the script

Time for BDF / Newton: 18.894 Time for BDF / Functional: 18.382 Time for Adams / Newton: 10.368 Time for Adams / Functional: 9.815 Time for Runge-Kutta: 4.743

These results show that on a nonstiff problem, for relatively same precision required and forcing the same step size, *Runge-Kutta* is faster.

Variable-size step ODE solvers are not appropriate for deterministic real-time applications because the computational overhead of taking a time step varies over the course of an application.

### See also

- LSodar — LSodar (short for Livermore Solver for Ordinary Differential equations, with Automatic method switching for stiff and nonstiff problems, and with Root-finding) is a numerical solver providing an efficient and stable method to solve Ordinary Differential Equations (ODEs) Initial Value Problems.
- CVode — CVode (short for C-language Variable-coefficients ODE solver) is a numerical solver providing an efficient and stable method to solve Ordinary Differential Equations (ODEs) Initial Value Problems. It uses either BDF or Adams as implicit integration method, and Newton or Functional iterations.
- IDA — Implicit Differential Algebraic equations system solver, providing an efficient and stable method to solve Differential Algebraic Equations system (DAEs) Initial Value Problems.
- Dormand-Prince 4(5) — Dormand-Prince is a numerical solver providing an efficient explicit method to solve Ordinary Differential Equations (ODEs) Initial Value Problems.
- Implicit Runge-Kutta 4(5) — Numerical solver providing an efficient and stable implicit method to solve Ordinary Differential Equations (ODEs) Initial Value Problems.
- Crank-Nicolson 2(3) — Crank-Nicolson is a numerical solver based on the Runge-Kutta scheme providing an efficient and stable implicit method to solve Ordinary Differential Equations (ODEs) Initial Value Problems. Called by xcos.
- DDaskr — Double-precision Differential Algebraic equations system Solver with Krylov method and Rootfinding: numerical solver providing an efficient and stable method to solve Differential Algebraic Equations systems (DAEs) Initial Value Problems
- Comparisons — This page compares solvers to determine which one is best fitted for the studied problem.
- ode — ordinary differential equation solver
- ode_discrete — ordinary differential equation solver, discrete time simulation
- ode_root — ordinary differential equation solver with roots finding
- odedc — discrete/continuous ode solver
- impl — differential algebraic equation

### Bibliography

### History

Version | Description |

5.4.1 | Runge-Kutta 4(5) solver added |

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