"DoubleStep" Method for NDSolve
Introduction
The method
performs a single application of Richardson's extrapolation for any one-step integration method.
Although it is not always optimal, it is a general scheme for equipping a method with an error estimate (hence adaptivity in the step size) and extrapolating to increase the order of local accuracy.
is a special case of extrapolation but has been implemented as a separate method for efficiency.
Given a method of order
:
- Take a step of size
to get a solution 
.
- Take two steps of size
to get a solution 
.
- Find an error estimate of order
as:
- The correction term
can be used for error estimation enabling an adaptive step-size scheme for any base method.
- Either use
for the new solution, or form an improved approximation using local extrapolation as:
- If the base numerical integration method is symmetric, then the improved approximation has order
; otherwise it has order 
.
Examples
Load some package with example problems and utility functions.
Select a nonstiff problem from the package.
Select a stiff problem from the package.
Extending Built-in Methods
The method
carries out one integration step using Euler's method. It has no local error control and hence uses fixed step sizes.
This integrates a differential system using one application of Richardson's extrapolation (see (
1)) with the base method
.
The local error estimate (
2) is used to dynamically adjust the step size throughout the integration.
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This illustrates how the step size varies during the numerical integration.
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An alternative base method is more appropriate for this problem.
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User-Defined Methods and Method Properties
Integration methods can be added to the
NDSolve framework.
In order for these to work like built-in methods it can be necessary to specify various method properties. These properties can then be used by other methods to build up compound integrators.
Method properties used by
are now described.
Order and Symmetry
This attempts to integrate a system using one application of Richardson's extrapolation based on the classical Runge-Kutta method.
Without knowing the order of the base method,
is unable to carry out Richardson's extrapolation.
This defines a method property to communicate to the framework that the classical Runge-Kutta method has order four.
The method
is now able to ascertain that
is of order four and can use this information when refining the solution and estimating the local error.
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The order of the result of Richardson's extrapolation depends on whether the extrapolated method has a local error expansion in powers of
or
(the latter occurs if the base method is symmetric).
If no method property for symmetry is defined, the
method assumes by default that the base integrator is not symmetric.
This explicitly specifies that the classical Runge-Kutta method is not symmetric using the
property.
Stiffness Detection
Details of the scheme used for stiffness detection can be found within
"StiffnessTest Method Option for NDSolve".
Stiffness detection relies on knowledge of the linear stability boundary of the method, which has not been defined.
Computing the exact linear stability boundary of a method under extrapolation can be quite complicated. Therefore a default value is selected which works for all methods. This corresponds to considering the
order power series approximation to the exponential at 0 and ignoring higher order terms.
- If
is True then a generic value is selected corresponding to a method of order 
(symmetric) or 
.
- If
is False then the property 
of the base method is checked. If no value has been specified then a default for a method of order 
is selected.
This computes the linear stability boundary for a generic method of order 4.
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A default value for the
property is used.
This shows how to specify the linear stability boundary of the method for the framework. This value will only be used if
is invoked with
.
assumes by default that a method is not appropriate for stiff problems (and hence uses stiffness detection) when no
property is specified. This shows how to define the property.
Higher Order
The following example extrapolates the classical Runge-Kutta method of order four using two applications of (
3).
The inner specification of
constructs a method of order five.
A second application of
is used to obtain a method of order six, which uses adaptive step sizes.
Nested applications of
are used to raise the order and provide an adaptive step-size estimate.
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In general the method
is more appropriate for constructing high-order integration schemes from low-order methods.
Option Summary
| | |
| "LocalExtrapolation" | True | specify whether to advance the solution using local extrapolation according to (4) |
| Method | None | specify the method to use as the base integration scheme |
| "StepSizeRatioBounds" | { ,4} | specify the bounds on a relative change in the new step size  from the current step size  as low ≤    ≤ high |
| "StepSizeSafetyFactors" | Automatic | specify the safety factors to incorporate into the error estimate (5) used for adaptive step sizes |
| "StiffnessTest" | Automatic | specify whether to use the stiffness detection capability |
Options of the method 
.
The default setting of
Automatic for the option
indicates that the stiffness test is activated if a nonstiff base method is used.
The default setting of
Automatic for the option
uses the values
for a stiff base method and
for a nonstiff base method.
method is restricted by stability rather than local accuracy.
