jabble
Class RungeKutta

java.lang.Object
  jabble.Evolver
      jabble.SingleLoopEvolver
          jabble.RungeKutta

public abstract class RungeKutta
extends SingleLoopEvolver

Solves a 1D Ordinary Differential Equation using Runge-Kutta.

Runge-Kutta is a method of numerically integrate an ordinary differential equation by using a step at the midpoint of an interval to cancel out lower-order error terms. We use the fourth order formula, sometimes refered as RK4. Given an ODE dy/dx = f(x, y), the algorithm is as follow:

k₁ = h f(x_n , y_n)
k₂ = h f(x_n + h/2, y_n + k₁/2)
k₃ = h f(x_n + h/2, y_n + k₂/2)
k₄ = h f(x_n + h, y_n+ k₃)
y_n+1 = y_n + k₁/6 + k₂/3 + k₃/3 + k₄/6 + O(h⁵)

where:

• h is the grid step size (i.e. dx)
• x_n and y_n are the values of x and y at the n^th point of the grid

In the Jabble framework, to use RungeKutta you must be working on a 1D grid. You need to create a class that extends RungeKutta, and declare the f() method, which corresponds to the f function. Instead of using the standard Field.at(Point) to calculate the value at a point, you should use the valueOf(Field): this way you can construct an f() method that calculates the function at the midpoint without doing so explicitely.

TODO: add example

Field Summary

Fields inherited from class jabble.Evolver

constantsMap, dDirs, fieldArrayMap, fieldMap, grid, stopTrigger

Constructor Summary

RungeKutta(java.lang.String fieldName)
Creates a new instance of RungeKutta

Method Summary

protected void	calculateNewFields(Point point) Calculates the value of the fields at the given Point.
protected abstract double	f() Computes f(x, y), the function that defines the ODE to be solved.
protected double	f1() Computes k1.
protected double	f2() Computes k2.
protected double	f3() Computes k3.
protected double	f4() Computes k4.
void	solve(Field field) Solves the ODE over the given Field.
void	solveAtPoint(Field field, Point point) Solves the ODE over the given Field at the given Point.
protected double	valueOf(Field field) Returns the value of the Field at the current Point, or mid-Point.

Methods inherited from class jabble.SingleLoopEvolver

calculateNewFields

Methods inherited from class jabble.Evolver

evolve, evolve, evolve, getNPreviousSlicesNeeded, initializeFieldArray, initializeFieldArray, skipBoundaryPoint

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

RungeKutta

public RungeKutta(java.lang.String fieldName)

Creates a new instance of RungeKutta

Method Detail

valueOf

protected final double valueOf(Field field)

Returns the value of the Field at the current Point, or mid-Point. This method is intended to be used within f() instead of Field.at(Point): for example, instead of double phiValue = phi.at(point); one would use double phiValue = valueOf(phi);.

Parameters:

field - a Field used during the computation

Returns:

the value of the field at the current Point (or mid-Point)

f

protected abstract double f()

Computes f(x, y), the function that defines the ODE to be solved. The function is not passed parameters. To compute the value of the different fields use valueOf(Field).

Returns:

value of f(x, y)

f1

protected double f1()

Computes k₁.

Returns:

value of k₁.

f2

protected double f2()

Computes k₂.

Returns:

value of k₂.

f3

protected double f3()

Computes k₃.

Returns:

value of k₃.

f4

protected double f4()

Computes k₄.

Returns:

value of k₄.

calculateNewFields

protected void calculateNewFields(Point point)

Description copied from class: SingleLoopEvolver

Calculates the value of the fields at the given Point.

Specified by:

calculateNewFields in class SingleLoopEvolver

Parameters:

point - the point where to compute the new value of the fields

solveAtPoint

public void solveAtPoint(Field field,
                         Point point)

Solves the ODE over the given Field at the given Point. Used in those cases in which one needs to solve more than one ODE.

Parameters:

field - Field to store the solution of the equation

point - Point at which to calculate the value

solve

public void solve(Field field)

Solves the ODE over the given Field. Used in those cases in which one needs to solve more than one ODE.

Parameters:

field - Field to store the solution of the equation