Implement Validation Tests for Integrators

[leebissessar5]

We have successfully encapsulated the Euler, Heun, and RK4 methods within our Integrator class. The unit testing framework we've constructed provides a robust preliminary examination of these methods across a set of three ODEs, benchmarking them against known solutions. However, this current setup predominantly confirms the capability of these solvers in handling ODEs, but it falls short of affirming the correct implementation of each individual integrator method.

To guarantee the correct implementation of our integrators, we need to add validation tests specific to each method. Here are some suggestions:

1. Analytical Comparisons:

   • For simple ODEs with known analytical solutions, compare the numerical results from every integrator with their exact solutions.
   • Calculate and document errors such as absolute or relative error for various step sizes.

2. Method-Specific Properties:

   • Confirm Euler's first-order accuracy, Heun's second-order accuracy, and RK4's fourth-order accuracy.
   • Also, test stability properties under different conditions.

3. Comparisons with Established Libraries:

   • Contrast the solutions derived from our integrators with those obtained from well-regarded numerical libraries.
   • Document any discrepancies and address significant deviations that arise.

This initiative aims to provide a more meticulous validation of our integrators, ensuring that each method is accurately implemented and performs reliably under a broad spectrum of conditions. Your insights and contributions towards this goal are highly valued.

Looking forward to your constructive feedback and collaboration on this endeavor.