Floating Point
An IEEE 754 floating-point number is made up of these three named fields.
What are the sign, exponent, and mantissa?
This is what must be done so that functions like solve_ivp can solve second-order ODE like Newton's second law .
What is reducing it to a system of coupled first-order ODEs?
Unlike Euler's method, the Verlet algorithm preserves this geometric property of Hamiltonian phase space.
What is symplecticity?
This is why most decimal fractions like 0.1 cannot be stored exactly in a computer.
What is that computers store numbers in base 2, not base 10?
This is the unphysical behavior you will observe in a long orbital simulation when using Euler's method.
What is energy drift?
In the Leapfrog algorithm, this quantity is computed at half-integer time steps, staggered relative to positions.
In numerical computing, these are the two distinct sources of error: one comes from the hardware's finite representation of numbers, the other from approximating a mathematical operation with a finite number of steps.
What are roundoff error and truncation error?
This is the general name used to refer to ODE solvers, there is a second and fourth order version.
What are the Runge-Kutta Methods?
Beeman's method improves velocity accuracy over Velocity Verlet by incorporating acceleration from this additional time level.
What is the previous time step?
In IEEE 754, all exponent bits set to 1 with a nonzero mantissa represents this
What is NaN (Not a Number)?
This is the term for ODEs that contain both very fast and very slow dynamics, causing explicit methods to require impractically tiny step sizes.
What are stiff equations?
This is the key practical disadvantage of position Verlet that both Velocity Verlet and Leapfrog were designed to fix.
What is the lack of explicit velocities at each time step?