Class DormandPrince853Integrator

All Implemented Interfaces:
FirstOrderIntegrator, ODEIntegrator

public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator
This class implements the 8(5,3) Dormand-Prince integrator for Ordinary Differential Equations.

This integrator is an embedded Runge-Kutta integrator of order 8(5,3) used in local extrapolation mode (i.e. the solution is computed using the high order formula) with stepsize control (and automatic step initialization) and continuous output. This method uses 12 functions evaluations per step for integration and 4 evaluations for interpolation. However, since the first interpolation evaluation is the same as the first integration evaluation of the next step, we have included it in the integrator rather than in the interpolator and specified the method was an fsal. Hence, despite we have 13 stages here, the cost is really 12 evaluations per step even if no interpolation is done, and the overcost of interpolation is only 3 evaluations.

This method is based on an 8(6) method by Dormand and Prince (i.e. order 8 for the integration and order 6 for error estimation) modified by Hairer and Wanner to use a 5th order error estimator with 3rd order correction. This modification was introduced because the original method failed in some cases (wrong steps can be accepted when step size is too large, for example in the Brusselator problem) and also had severe difficulties when applied to problems with discontinuities. This modification is explained in the second edition of the first volume (Nonstiff Problems) of the reference book by Hairer, Norsett and Wanner: Solving Ordinary Differential Equations (Springer-Verlag, ISBN 3-540-56670-8).

Since:
1.2
  • Field Details

    • METHOD_NAME

      private static final String METHOD_NAME
      Integrator method name.
      See Also:
    • STATIC_C

      private static final double[] STATIC_C
      Time steps Butcher array.
    • STATIC_A

      private static final double[][] STATIC_A
      Internal weights Butcher array.
    • STATIC_B

      private static final double[] STATIC_B
      Propagation weights Butcher array.
    • E1_01

      private static final double E1_01
      First error weights array, element 1.
      See Also:
    • E1_06

      private static final double E1_06
      First error weights array, element 6.
      See Also:
    • E1_07

      private static final double E1_07
      First error weights array, element 7.
      See Also:
    • E1_08

      private static final double E1_08
      First error weights array, element 8.
      See Also:
    • E1_09

      private static final double E1_09
      First error weights array, element 9.
      See Also:
    • E1_10

      private static final double E1_10
      First error weights array, element 10.
      See Also:
    • E1_11

      private static final double E1_11
      First error weights array, element 11.
      See Also:
    • E1_12

      private static final double E1_12
      First error weights array, element 12.
      See Also:
    • E2_01

      private static final double E2_01
      Second error weights array, element 1.
      See Also:
    • E2_06

      private static final double E2_06
      Second error weights array, element 6.
      See Also:
    • E2_07

      private static final double E2_07
      Second error weights array, element 7.
      See Also:
    • E2_08

      private static final double E2_08
      Second error weights array, element 8.
      See Also:
    • E2_09

      private static final double E2_09
      Second error weights array, element 9.
      See Also:
    • E2_10

      private static final double E2_10
      Second error weights array, element 10.
      See Also:
    • E2_11

      private static final double E2_11
      Second error weights array, element 11.
      See Also:
    • E2_12

      private static final double E2_12
      Second error weights array, element 12.
      See Also:
  • Constructor Details

    • DormandPrince853Integrator

      public DormandPrince853Integrator(double minStep, double maxStep, double scalAbsoluteTolerance, double scalRelativeTolerance)
      Simple constructor. Build an eighth order Dormand-Prince integrator with the given step bounds
      Parameters:
      minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
      maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
      scalAbsoluteTolerance - allowed absolute error
      scalRelativeTolerance - allowed relative error
    • DormandPrince853Integrator

      public DormandPrince853Integrator(double minStep, double maxStep, double[] vecAbsoluteTolerance, double[] vecRelativeTolerance)
      Simple constructor. Build an eighth order Dormand-Prince integrator with the given step bounds
      Parameters:
      minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
      maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
      vecAbsoluteTolerance - allowed absolute error
      vecRelativeTolerance - allowed relative error
  • Method Details

    • getOrder

      public int getOrder()
      Get the order of the method.
      Specified by:
      getOrder in class EmbeddedRungeKuttaIntegrator
      Returns:
      order of the method
    • estimateError

      protected double estimateError(double[][] yDotK, double[] y0, double[] y1, double h)
      Compute the error ratio.
      Specified by:
      estimateError in class EmbeddedRungeKuttaIntegrator
      Parameters:
      yDotK - derivatives computed during the first stages
      y0 - estimate of the step at the start of the step
      y1 - estimate of the step at the end of the step
      h - current step
      Returns:
      error ratio, greater than 1 if step should be rejected