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Testing Methods: Solving Initial Value Problems

In addition to directly analyzing solver properties, NodePy also facilitates the testing of solvers through application to problems of interest. Furthermore, NodePy includes routines for automatically running sets of tests to compare the practical convergence or efficiency of various solvers for given problem(s).

Initial Value Problems

The principal objects in NodePy are ODE solvers. The object upon which a solver acts is an initial value problem. Mathematically, an initial value problem (IVP) consists of one or more ordinary differential equations and an initial condition:

\begin{align*} u’(t) & = F(u) & u(0) & = u_0. \end{align*}

In NodePy, an initial value problem is an object with the following properties:

  • rhs(): The right-hand-side function; i.e. F where \(u(t)'=F(u)\).

  • u0: The initial condition.

  • T: The (default) final time of solution.

Optionally an IVP may possess the following:
  • exact(): a function that takes one argument (t) and returns the exact solution (Should we make this a function of u0 as well?)

  • dt0: The default initial timestep when a variable step size integrator is used.

  • Any other problem-specific parameters.

The module ivp contains functions for loading a variety of initial value problems. For instance, the van der Pol oscillator problem can be loaded as follows:

>> from NodePy import ivp
>> myivp = ivp.load_ivp('vdp')

Solving Initial Value Problems

Any ODE solver object in NodePy can be used to solve an initial value problem simply by calling the solver with an initial value problem object as argument:

>> t,u = rk44(my_ivp)

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