nintegration

Intro

Numerical Integration of a system of de's (Differential Equations).

Integrating a system of de's (differential equations) is important in modeling systems and hence simulating physical events.

The DEsys is a very general numerical integration of de's class. It relies on the client configuring the data and functions, indeed the interface needs close examination.

The class was generalized to higher dimensions making it useful. For systems with greater than one dimension the infinitesimals will need to see other system variables, which are public to DEsys.

Letting the client define there own integration methods is really versatile, and if you want just a good one then use RK for "plug and play".

• Tested on a second order de and a two variable system.
• Templated component architecture. Virtual functional objects used.

Source

Files

Doxygen

What is an Infinitesimal

Something very small, a magnitude smaller than other numbers around.

Example

and
has the solution

By testing the integration methods against a known solution it could be determined whether the solution was being approximated.

The testing for a system of de's was done by transforming the above system into a coupled system.

Let then
and
and

History

$ ./main prog=5 xmax=50 sample=100 h=0.0001 $ gnuplot gnup002.txt

Equation Ordering

Does the ordering of equations influence the solution? I conjectured that it did but the example de gave interesting results.

The de solver is an order driven system (the first dimension is solved, then the second, ... ) its pretty accurate though its not the sort of thing discussed in a maths degree.

For example if you convert a single variable de to a system of de's, what ordering (if any) of variables has the highest accuracy?

I solved the example de 3 ways. The first two were as given in the example where the second order de was reduced to two equations. These first order equations were solved.

The third way was as one second order de with the first derivative integrated and other derivatives having Euler method applied. Since Euler's method was being used as the integrator this was exactly the same as one of the previous ways. It differed not in results but implementation.

What is surprising about these results is that if you sum the errors you get a number 3 orders of magnitude less. You do not have to be Einstein to realize that summing the two solutions and taking the average will give you possibly 3 orders of magnitude more accuracy for double the cost.

TODO
This is a gain. If you incorporate it into a new integrator it would yield one more order of magnitude to give you 4. So for this particular de the order of equations can be used. However for 3 equations there are 3!=6 possible orderings and another approach may be better. Also is this some property of the particular equation under investigation?

Lets assume that this can be generalized so that for n equations there are n! ways of integrating. Maybe combinations or other things could be used to reduce this a bit. Now assume an ideal parallel computer/s. The algorithm could be done in parallel.

The idea could be used to write new approximators locally as I have been inferring. The experiment used separate integrators. Is 100 places of accuracy possible?

TODO

• Higher Order Approximators
  Interpolate the second derivative. y[n+1] = y[n] + hf(x[n],y[n]) + h^2/2*g(x[n],y[n],dy[n]))
• Macro of 1D k'th order de's
  Let client just put in class name and function into macro.
• Orbiting planet
  how long before EL method breaks down, how many iterations before RK breaks down - no longer sinusoidal behavior.
• Physics problems
  As time evolves use this class in simulation and de calculation.
  non-linear pendulum simulation
  springs
  wave equation
  visualize by reproducing the data or plotting the direction field.