## Monday, June 10, 2013

### Runge-Kutta ODE solver in Scala

This post describes the implementation of the different Runge-Kutta method to solve differential equations in Scala.
The objective is to leverage the functional programming components of the Scala programming language to create a generic solver of ordinary differential equations (ODE) using Runge-Kutta family of approximation algorithms.

Overview
Most of ordinary differential equations cannot be solved analytically. In this case, a numeric approximation to the solution is often good enough to solve an engineering problem. Oddly enough most of commonly used algorithm to compute such an approximation have been establish a century ago. Let's consider the differential equation $\frac{\mathrm{d}y }{\mathrm{d} x} = f(x,y)$ The family of explicit Runge-Kutta numerical approximation methods is defined as $y_{n+1} = y_{n} + \sum_{i=0}^{s<n}b_{i}k_{i}\\where\,\,k_{j}=h.f(x_{n} + c_{j}h, y_{n} + \sum_{s=1}^{j-1} a_{s,s-1}k_{s-1} )\\with\,\,h=x_{n+1}-x_{n}\,\,and\,\,\Delta = \frac{dy}{dx}+ \sum_{s=1}^{j-1} a_{s,s-1}k_{s-1}$ k(j) is the increment based on the slope at the midpoint of the interval [x(n),x(n+1)] using delta. The Euler method defined as $y_{n+1} = y_{n} + hf(t_{n},y_{n})$ and 4th order Runge-Kutta $y_{n+1} = y_{n} + \frac{h}{6}(k_{1} + 2 k_{2}+2 k_{3}+ k_{4})\,\,\,;h = x_{n+1}-x_{n}\\k_{1}=f(x_{n},y_{n})\\k_{2}=f(x_{n}+\frac{h}{2}, y_{n}+\frac{hk_{1}}{2})\\k_{3}=f(x_{n}+\frac{h}{2},y_{n}+\frac{hk_{2}}{2})\\k_{4}=f(x_{n}+h,y_{n}+hk_{3})$

The implementation relies on the functional aspect of the Scala language and should be flexible enough to support any new future algorithm. The generic Runge-Kutta coefficients a(i), b(i) and c(i) are represented as a matrix: $\begin{vmatrix} c_{2}\,\,a_{21}\,\,0.0\,\,0.0\,\,...\,\,0.0\,\,0.0\\ c_{3}\,\,a_{31}\,\,a_{32}\,\,0.0\,\,...\,\,0.0\,\,0.0 \\ c_{4}\,\,a_{41}\,\,a_{42}\,\,a_{43}\,\,...\,\,0.0\,\,0.0\\ \\ c_{i}\,\,a_{i1}\,\,a_{i2}\,\,a_{i3}\,\,...\,\,\,\,...\,\,a_{ii-1}\\*\,\,b_{1}\,\,b_{2}\,\,b_{3}\,\,...\,\,...\,\,b_{i-1}\,\,b_{i} \end{vmatrix}$ In order to illustrate the flexibility of this implementation using Scala, I encapsulate the matrix of coefficients of the Euler, 3th order Runge-Kutta, 4th order Runge-Kutta and Felhberg methods using enumeration and case classes.

Note: For the sake of readability of the implementation of algorithms, all non-essential code such as error checking, comments, exception, validation of class and method arguments, scoping qualifiers or import is omitted

Coefficients": Enumerators and case classes
Java developers, are familiar with enumerators as a data structure to list values without the need to instantiate an iterable collection.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 trait RungeKuttaCoefs { type COEFS = Array[Array[Double]] val EULER = Array( Array[Double](0.0, 1.0) ) // Coefficients for Runge-Kutta of order 3 val RK3 = Array( Array[Double](0.0, 0.0, 1/3, 0.0, 0,0), Array[Double](0.5, 0.5, 0.0, 2/3, 0.0), Array[Double](1.0, 0.0, -1.0, 0.0, 1/3) ) // Coefficients for Runge-Kutta of order 4 val RK4 = Array( Array[Double](0.0, 0.0, 1/6, 0.0, 0,0, 0.0), Array[Double](0.5, 0.5, 0.0, 1/3, 0.0, 0.0 ), Array[Double](0.5, 0.0, 0.5, 0.0, 1/3, 0.0), Array[Double](1.0, 0.0, 0.0, 1.0, 0.0, 1/6) ) // Coefficients for Runge-Kutta/Felberg of order 5 val FELBERG = Array( Array[Double](0.0, 0.0, 25/216, 0.0, 0.0, 0.0, 0.0, 0.0), Array[Double](0.25, 0.25, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ), Array[Double](3/8, 3/32, 0.0, 0.0, 1408/2565, 0.0, 0.0, 0.0), Array[Double](12/13,1932/2197,-7200/2197, 7296/2197, 0.0,2197/4101,0.0,0.0), Array[Double](1.0, 439/216, -8.0, 3680/513, -845/4104, 0.0, -1/5, 0.0), Array[Double](0.5, -8/27, 2.0, -3544/2565, 1859/4104, -11/40, 0.0, 0.0) ) val rk = List[COEFS](EULER, RK3, RK4, FELBERG) } object RungeKuttaForms extends Enumeration with RungeKuttaCoefs{ type RungeKuttaForms = Value val Euler, Rk3, Rk4, Fehlberg = Value @inline final def getRk(value: Value): COEFS = rk(value.id) } 

The first step is to encapsulate the coefficients of the various versions of the Runge-Kutta formuler (lines 4 & 5), Runge-Kutta order 3 (lines 9 - 12), Runge-Kutta order 4 (lines 16 - 20) and Runge-Kutta-Felberg order 5 (lines 24 - 30) in an Scala enumerator.

The enumerator is at best not elegant. As you browse throught the code snippet above, it is clear that the design to wrap the matrices of coefficients with the enumerator is quite cumbersome. There is a better way: pattern matching Case classes could be used instead of the singleton enumeration. Setters or getters can optionally be added as in the example below.
The validation of the arguments of methods, exceptions and auxiliary method or variables are omitted for the sake of simplicity.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 trait RKMethods { type COEFS = Array[Array[Double]] def getRk(i: Int, j: Int): COEFS object class Euler extends RKMethods{ Array(Array[Double](0.0, 1.0)) override def getRk(i: Int, j: Int): COEFS {} } object class RK3 extends RKMethods { Array( Array[Double](0.0, 0.0, 1/3, 0.0, 0,0), Array[Double](0.5, 0.5, 0.0, 2/3, 0.0), Array[Double](1.0, 0.0, -1.0, 0.0, 1/3) ) override def getRk(i: Int, j: Int): COEFS {} } .... 

The first step is to encapsulate the coefficients of the various versions of the Runge-Kutta formuler (lines 4 & 5), Runge-Kutta order 3 (lines 9 - 12), Runge-Kutta order 4 (lines 16 - 20) and Runge-Kutta-Felberg order 5 (lines 24 - 30) in an Scala enumerator.

In this second approach, the values of the enumerator are replaced y Euler object (line 5), RK3 - Runge-Kutta order 3 (line 10).

Integration
The main class RungeKutta implements all the components necessary to resolve the ordinary differential equation. This simplified implementation relies on adjustable step for the integration.

  1 2 3 4 5 6 7 8 9 10 11 12 class RungeKutta( rungeKutta: RungeKuttaForm, adjustStep: (Double, AdjustParameters) => (Double), adjustParameters: AdjustParameters) { final class StepIntegration(val coefs: Array[Array[Double]]) {} def solve( xBegin: Double, xEnd: Double, derivative: (Double, Double) => Double): Double } 

The class RungeKutta has three arguments
• rungeKutta form or type of the Runge-Kutta formula (line 2)
• adjustStep Metric function to adjust the integration step, dx (line 3)<.li>
• adjustParameters Parameters used to compute the derivative (line 4)

The computation of the parameters to adjust the integration step, in the code snippet below, is rather simple. A more elaborate implementation would include several alternative formulas implemented as sealed case class

 1 2 3 4 5 6 7 8 case class AdjustParameters( maxDerivativeValue: Double = 0.01, minDerivativeValue: Double = 0.00001, gamma: Double = 1.0) { lazy val dx0 = 2.0*gamma/(maxDerivativeValue + minDerivativeValue) } 

The sum of the previous Ks value is computed through an inner loop. The outer loop computes all the values for k using the Runge-Kutta matrix coefficients for this particular method. The integration step is implemented as a tail recursion (lines 14 - 22). but an iterative methods using foldLeft can also be used. The tail recursion may not be as effective in this case because it is implemented as a closure: the method has to close on ks.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 final class StepIntegration(val coefs : Array[Array[Double]] ) { // Main routine def compute( x: Double, y: Double, dx: Double, derivative: (Double, Double) => Double): Double = { val ks = new Array[Double](coefs.length) // Tail recursion closure @scala.annotation.tailrec def compute(i: Int, k: Double, sum: Double): Double= { ks(i) = k val sumKs= (0 until i)./:(0.0)((s, j) => s + ks(j)*coefs(i)(j+1)) val newK = derivative(x + coefs(i)(0)*dx, y + sumKs*dx) if( i >= coefs.size) sum + newK*coefs(i)(i+2) else compute(i+1, newK, sum + newK*coefs(i)(i+2)) } dx*compute(0, 0.0, 0.0) } 

The next method implements the generic solver that iterates through the entire integration interval. As a matter of fact the solver is indeed implemented as a tail recursion (lines 8 - 20).
The accuracy of the solver depends on the value of the increment value dx as computed on line 17. We need to weight the accuracy provided by infinitesimal increment with its computation cost.  Ideally an adaptive algorithm that compute the value dx according the value dy/dx or delta would provide a good compromise between accuracy and cost. The recursion ends when the recursion value x reaches the end of the integration interval (line 14)..

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 def solve( xBegin: Double, xEnd: Double, derivative: (Double, Double) => Double): Double ={ val rungeKutta = new StepIntegration(rungeKuttaForm) @scala.annotation.tailrec def solve( x: Double, y: Double, dx: Double, sum: Double): Double = { val z = rungeKutta.compute(x, y, dx, derivative) if( x >= xEnd) sum + z else { val dx = adjustStep(z - y, adjustParameters) solve(x + dx, z, dx, sum+z) } } solve(xBegin, 0.0, adjustParameters.initial, 0.0) } 

The invocation of the solver is very straight forward and can be verified against the analytical solution.
The first step is to define the function that adjusts the integration step (lines 1, 10). This implementation uses the default adjust paramters (line 19) in the initialization of the solver (lines 16 - 19).

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 val adjustingStep = (diff: Double, adjustParams: AdjustParameters) => { val dx = Math.abs(diff)*adjustParams.dx0/adjustParams.gamma if( dx < adjustParams.minDerivativeValue) adjustParams.minDerivativeValue else if ( dx > adjustParams.maxDerivativeValue) adjustParams.maxDerivativeValue else dx } final val x0 = 0.0 final val xEnd = 2.0 val solver = new RungeKutta( Rk4, adjustingStep, AdjustParameters()) solver.solve( x0, xEnd, (x: Double, y: Double) => Math.exp(-x*x)) 

The family of explicit Runge-Kutta methods provides a good starting point to resolve ordinary differential equations. The current implementation could and possibly should be extended to support adaptive dx managed by a control loop using a reinforcement learning algorithm of a Kalman filter of just simple exponential moving average.

References

The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods - E. Hairer, C Lubich, M. Roche - Springer - 1989
Programming in Scala - M. Odersky, L. Spoon, B. Venners - Artima Press 2008
https://github.com/prnicolas