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ExplicitMidpoint.md

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Explicit Midpoint

Category

Second-order explicit Runge-Kutta method.

Implemented MATLAB file

src/methods/explicit_midpoint.m

Core idea

Second-order, two RHS evaluations, better accuracy than Euler.

The method advances an initial-value problem

y' = f(t, y),    y(t0) = y0

from t_n to t_(n+1)=t_n+h using the method-specific update formula implemented in the MATLAB file above.

Historical background

Second-order Runge-Kutta formulas were developed in the Runge-Kutta tradition initiated by Runge and Kutta around 1895-1901.

The documentation in this repository is intended as a practical engineering summary. For formal historical work, consult the primary references listed in ../references.md.

Strengths

  • Gives a clear benchmark representative of the Second-order explicit Runge-Kutta method family.
  • Useful for comparing error, runtime, stability, and invariant behavior.
  • Easy to inspect because the implementation is intentionally written in readable MATLAB.

Weaknesses and limitations

  • No single integrator is best for all differential equations.
  • Fixed-step methods require careful step-size selection.
  • Implicit methods require nonlinear or linear solves.
  • Symplectic and energy-oriented methods are meaningful mainly for mechanical/Hamiltonian problems.

Works best for

smooth non-stiff problems

Main performance metrics for this method

Useful metrics include:

  • final-time error,
  • maximum trajectory error,
  • RMS trajectory error,
  • CPU time,
  • number of right-hand-side evaluations,
  • rejected steps for adaptive variants,
  • Newton iterations for implicit variants,
  • energy or invariant drift when applicable,
  • failure rate on stiff or unstable tests.

Interpretation in this repository

In the generated benchmark plots, this method should be interpreted by problem class:

  • On non-stiff equations, compare error per CPU time.
  • On stiff equations, check whether the method survives without tiny step sizes.
  • On conservative mechanics, check energy/invariant drift rather than only pointwise error.
  • On chaotic equations, use short-time error and long-time qualitative behavior.

References

See ../references.md.