Numerical Methods

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This section covers the numerical algorithms that power the Apogee simulator.

LaTeX Reference

This section corresponds to Sections 6 and 8 of nm_final_project.tex.

Overview

The simulator combines three families of numerical methods:

  1. ODE Integration - Solving the differential equations
  2. Optimization - Finding optimal control parameters
  3. Numerical Stability - Handling singularities and edge cases

Method Summary

Problem Method Package
Initial Value Problem Tsitouras 5(4) RK diffrax
Event Detection Newton-Raphson optimistix
Trajectory Optimization Levenberg-Marquardt Custom
Jacobian Approximation Finite Differences Custom
Jacobian Update Broyden Rank-1 Custom
Bounded Optimization Logistic Transform Custom

Topics

ODE Integration

Solving the equations of motion:

\[ \frac{d\mathbf{y}}{dt} = \mathbf{f}(\mathbf{y}, t) \]

Optimization

Finding optimal control parameters:

\[ \min_{\mathbf{u}} \|\mathbf{F}(\mathbf{u})\|^2 \]

Numerical Stability

Handling numerical challenges:

Algorithm Flow

graph TD
    A[User Input<br/>h_target, payload] --> B[Multistart<br/>Generate candidates]
    B --> C[Best Candidate]
    C --> D[Newton Iteration]
    D --> E{Converged?}
    E -->|No| F[Compute Jacobian]
    F --> G[LM Step]
    G --> H[Line Search]
    H --> I[Broyden Update]
    I --> D
    E -->|Yes| J[Optimal Solution]

    G --> K[Simulate Trajectory<br/>Tsit5 + Events]
    K --> L[Orbital Diagnostics]
    L --> M[Compute Residuals]
    M --> G

Convergence Characteristics

Typical Performance

Metric Value
Newton iterations 5-15
Function evaluations 20-50
Wall-clock time (first run) ~20 seconds
Wall-clock time (subsequent) ~0.5 seconds

The first run is slower due to JAX JIT compilation.

Accuracy

Metric Typical Value
Eccentricity < 0.001
Altitude error < 2 km
Velocity error < 2 m/s
FPA at insertion < 0.1°

Key Parameters

ODE Solver

rtol = 1e-6    # Relative tolerance
atol = 1e-6    # Absolute tolerance
dt0 = 0.5      # Initial step size [s]
max_steps = 100_000

Shooting Method

tol = 1e-6     # Convergence tolerance
max_iter = 30  # Maximum Newton iterations
max_evals = 200  # Maximum function evaluations

References

  1. Stoer, J. & Bulirsch, R. (2002). Introduction to Numerical Analysis. Springer. Chapters 5-7.

  2. Nocedal, J. & Wright, S. (2006). Numerical Optimization. Springer. Chapters 6 and 10.

  3. Hairer, E., Nørsett, S.P., & Wanner, G. (1993). Solving Ordinary Differential Equations I. Springer.

Next Steps

Flight Phases ODE Integration