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 Detail

      • METHOD_NAME

        private static final java.lang.String METHOD_NAME
        Integrator method name.
        See Also:
        Constant Field Values
      • 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:
        Constant Field Values
      • E1_06

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

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

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

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

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

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

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

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

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

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

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

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

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

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

        private static final double E2_12
        Second error weights array, element 12.
        See Also:
        Constant Field Values
    • Constructor Detail

      • 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 Detail

      • 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