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# Implicit Runge-Kutta 4(5)

*Implicit Runge-Kutta* is a numerical solver providing an efficient and stable implicit method to solve Ordinary Differential Equations (ODEs) Initial Value Problems. Called by xcos.

### Description

*Runge-Kutta* is a numerical solver providing an efficient and stable 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.

This method being implicit, it can be used on stiff problems.

It is an enhancement of the backward Euler method, which approximates
*y _{n+1}*
by computing

*f(t*and truncating the Taylor expansion.

_{n}+h, y_{n+1})The implemented scheme is inspired from the "Low-Dispersion Low-Dissipation Implicit Runge-Kutta Scheme" (see bottom for link).

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

Here, the next value is determined by the present value
*y _{n}*
plus the weighted average of three 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)*. They are distributed approximately equally on the interval.

*k1*is the increment based on the slope near the quarter of the interval, using*y*,_{n}+ a11*h*k1,*k2*is the increment based on the slope near the midpoint of the interval, using*y*,_{n}+ a21*h*k1 + a22*h*k2,*k3*is the increment based on the slope near the third quarter of the interval, using*y*_{n}+ a31*h*k1 + a32*h*k2 + a33*h*k3.

We see that the computation of *ki* requires *ki*, thus necessitating the use of a nonlinear solver (here, fixed-point iterations).

First, we set
*k0 = h * f(t _{n}, y_{n})*
as first guess for all the

*ki*, to get updated

*ki*and a first value for

*y*.

_{n+1}Next, we save and recompute
*y _{n+1}*
with those new

*ki*.

Then, we compare the two
*y _{n+1}*
and recompute it until its difference with the last computed one is inferior to the simulation parameter

*reltol*.

This process adds a significant computation time to the method, but greatly improves stability.

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

*k3*, we are approximating

*y*, thus making an error in

^{(3)}_{n}*O(h*. But choosing the right coefficients in the computation of the

^{4})*ki*(notably the

*a*) makes us gain an order, thus making a per-step total error in

_{ij}*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 *implicit 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 *implicit RK* 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 implicit RK and set the precision scs_m.props.tol(2) = 10^-10; scs_m.props.tol(6) = 7; 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(t, "Time for implicit Runge-Kutta:");

The Scilab console displays:

Time for implicit Runge-Kutta: 8.911

Now, in the following script, we compare the time difference between *implicit RK* 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 implicit Runge-Kutta: 6.652

These results show that on a nonstiff problem, for relatively same precision required and forcing the same step size, *implicit Runge-Kutta* competes with *Adams / Functional*.

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

### Bibliography

Journal of Computational Physics, Volume 233, January 2013, Pages 315-323 A low-dispersion and low-dissipation implicit Runge-Kutta scheme

### History

バージョン | 記述 |

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

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