jabble
Class IteratedCrankNicholson

java.lang.Object
  jabble.Evolver
      jabble.IteratedCrankNicholson

public abstract class IteratedCrankNicholson
extends Evolver

This class implements the Iterated Crank-Nicholson template algorithm. It provides the common iteration and averaging pattern for the algorithm, leaving out only the part relative to the specific equation.

Given a finite difference expression u_t+dt = u_t + dt f(u_t), where u_t is a field at the time t, u_t+dt is the same field at the following time step, and f the stencil used to calculate du, the Iterated Crank-Nicholson is defined as:

u_t+dt = ICN(u_t, dt, f, k):

1. n = 0; u = u_t;
   set the current iteration to 0 and initialize average to u_t.
2. u_t+dt = u_t + dt f(u);
   calculate solution using the average to compute the differential.
3. if (n == k) return u_d+dt;
   return the solution if all iterations are done.
4. u = (u_t + u_t+dt) / 2;
   calculate average between solutions at previous and following time steps.
5. n = n + 1;
   increase the iteration counter.
6. go to step 2.

where:

• k is the number of iterations. Typically this number is 2 as more iteration won't improve the solution.
• u is the average between the field given as input and the field calculated at the last iteration

In the Jabble framework, this class will take care of performing the iteration and the averaging. One needs to specify:

• the number of iterations k , which will be specified through the constructor. The default is 2.
• the fields on which the method should be performed. The Evolver will recognize the pattern of xxx, averageXxx and previousXxx as the fields on which ICN will take place, and the averages will be calculated automatically.
• the finite different expression by overriding the method { @link #calculateNewFields(Point) . The expression has to calculate the field as a function of the previousField and averageField as required by the method.

Notice that more than one field can be calculated at the same time.

The following is a sample code for an IteratedCrankNicholson class that solves the Advection equation dphi / dt = dphi / dx :

public class AdvectionCartesian1DICN extends IteratedCrankNicholson {

     // Constant representing the X direction
     private static final int X = 0;

     // The fields on which to use the ICN method
     private Field previousPhi;
     private Field phi;
     private Field averagePhi;

     // The time differential
     private Field dt;

     // The finite difference expression
     //used during the iterations
     protected void calculateNewFields(Point point) {
         double dtValue = dt.at(point);
         double phiValue = previousPhi.at(point);
         phiValue += dtValue * centeredDifference(averagePhi, X);
         phi.setAt(point,phiValue);
     }
   
 }

Reference:

• S. Teukolsky Phys.Rev.D 61:087501,2000 gr-qc/990926

Field Summary

Fields inherited from class jabble.Evolver

constantsMap, dDirs, fieldArrayMap, fieldMap, grid, stopTrigger

Constructor Summary

protected	IteratedCrankNicholson() Creates a new Iterated Crank-Nicholson evolver, with the default of 2 iterations.
protected	IteratedCrankNicholson(int nIterations) Creates a new Iterated Crank-Nicholson evolver, specifying the number of iterations.

Method Summary

protected void	calculateNewFields() Implements the skeleton of the template algorythm.
protected abstract void	calculateNewFields(Point point) Calculate the new value of the fields at the given point.

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

IteratedCrankNicholson

protected IteratedCrankNicholson(int nIterations)

Creates a new Iterated Crank-Nicholson evolver, specifying the number of iterations.

Parameters:

nIterations - number of iterations for the Iterated Crank-Nicholson algorithm

IteratedCrankNicholson

protected IteratedCrankNicholson()

Creates a new Iterated Crank-Nicholson evolver, with the default of 2 iterations.

Method Detail

calculateNewFields

protected abstract void calculateNewFields(Point point)

Calculate the new value of the fields at the given point. One should calculate a field (phi) given the value at the previous time step (previousPhi) and the increment as a function of the ICN averaged field (averagePhi).

Parameters:

point - the point at which to calculate the value of the new fields.

calculateNewFields

protected final void calculateNewFields()

Implements the skeleton of the template algorythm. To specialize it for a particular set of equations, override calculateNewFields(Point)

Specified by:

calculateNewFields in class Evolver