ODE fundamentals, Euler & Runge-Kutta

What is an ODE?

Equation involving derivatives of a function

What determines the order of an ODE?

Highest derivative present

What is a general solution?

Contains constants

What is a particular solution?

Specific solution after applying initial conditions

What is an initial value problem (IVP)?

ODE + initial condition(s)

How many initial conditions are needed?

1st order → 1, 2nd order → 2

What type of method is Euler’s method?

First-order numerical method

Euler method error scales as:

∼Δx

What is the main weakness of Euler’s method?

Low accuracy

What does RK2 improve?

Accuracy using multiple slope evaluations

RK2 requires how many function evaluations?

Two

Local truncation error is:

Error from one step of approximation

Global error is:

Accumulated error over all steps