|
DoubleStep IntroductionThe method DoubleStep 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. DoubleStep 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  . ExamplesLoad some package with example problems and utility functions. In[5]:=  |
Select a nonstiff problem from the package. In[7]:=  |
Select a stiff problem from the package. In[8]:=  |
Extending built-in methodsThe method ExplicitEuler 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 ExplicitEuler. The local error estimate (-1) is used to dynamically adjust the step size throughout the integration. In[9]:=  |
Out[9]=
|
This illustrates how the step size varies during the numerical integration. In[10]:=  |
Out[10]=
|
The stiffness detection device (described within StiffnessTest) ascertains that the ExplicitEuler method is restricted by stability rather than local accuracy. In[11]:=  |
Out[11]=
|
An alternative base method is more appropriate for this problem. In[12]:=  |
Out[12]=
|
User-defined methods and method propertiesIntegration 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. Here is how to define a top-level plug-in for the classical Runge-Kutta method (see ExplicitRungeKutta for more details). In[13]:=  |
Method properties used by DoubleStep are now described. Order and symmetryThis attempts to integrate a system using one application of Richardson's extrapolation based on the classical Runge-Kutta method. In[14]:=  |
Without knowing the order of the base method, DoubleStep 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. In[15]:=  |
The method DoubleStep is now able to ascertain that ClassicalRungeKutta is of order four and can use this information when refining the solution and estimating the local error In[16]:=  |
Out[16]=
|
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 DoubleStep 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 SymmetricMethodQ property. In[17]:=  |
Stiffness detectionStiffness detection relies on knowledge of the linear stability boundary of the method, which has not been defined. In[18]:=  |
This gives the linear stability boundary of the method for the framework. In[19]:=  |
The scheme is now equipped with stiffness detection. In[20]:=  |
Out[20]=
|
DoubleStep assumes by default that a method is not appropriate for stiff problems (and hence uses stiffness detection) when no StiffMethodQ property is specified. This shows how to define the property. In[21]:=  |
Higher orderThe following example extrapolates the classical Runge-Kutta method of order four using two applications of (-1). The inner specification of DoubleStep constructs a method of order five. A second application of DoubleStep is used to obtain a method of order-six, which uses adaptive step sizes. Nested applications of DoubleStep are used to raise the order and provide and adaptive step size estimate. In[22]:=  |
Out[22]=
|
In general the method Extrapolation is more appropriate for constructing high-order integration schemes from low-order methods. Option summaryOptions of the method DoubleStep.
| 
