Euler's Method

euler, differential equation, euler explicit method, set of odes, basic

A thorough walkthrough of the theoretical aspects of Euler's method. Also covers how to solve higher order ODEs.

  Simple implementation of Euler's method

explicit euler method, ode, implementation, basic

Basic notebook covering how to implement Euler's method, without much focus on theory

  Uniform Magnetic Field

magnetism, explicit euler method, ode, Lorentz' law

Solving Lorentz force law for a charged particle traveling in a uniform magnetic field using Euler's method.

  Projectile motion

explicit euler method, 4th order runge-kutta, ode, Big Bertha, set of odes

Computing the trajectory of a projectile moving through the air, subject to wind and air drag.

  Gravity Assist

animation, gravity, newton, semi-implicit euler method

Explaining the concept and simulating gravitational slingshot of a spacecraft passing a planet.

  Double Pendulum and Chaos

animation, differential equation, Lagrangian, Euler-Lagrange equations, chaos, phase space, odeint

Discusses the chaotic motion of the double pendulum using a phase-space diagram

  Simple Pendulum

animation, explicit euler method, ode

Simulates the simple pendulum and damped simple pendulum

  Euler Equations for Inviscid Flow

An introduction to the compressible Euler equations and methods for solving them numerically.

  Sloshing

simpson's method, newton, euler, eigenvalue, poisson's equation, integration, interpolation

Analyzing sloshing using a numerical approach based on a linear model, which reduces the problem to a Steklov eigenvalue problem.

  Planetary Motion

space, explicit euler method, gravity, newton

Applying the explicit and implicit Euler methods and the fourth order Runge-Kutta method to calculate the trajectory of the Earth around the Sun.

  Partial Differential Equations - Two Examples

animation, laplace's equation, finite-differences, pde, differential equation, stability, implicit euler method

This module shows two examples of how to discretize partial differential equations: the 2D Laplace equation and 1D heat equation.

  Adaptive Runge-Kutta Methods

explicit euler method, 4th order runge-kutta, embedded runge-kutta pair, trapezoidal method, ode

Solving a first-order ordinary differential equation using Runge-Kutta methods with adaptive step sizes.

  Runge-Kutta Methods

explicit euler method, 4th order runge-kutta, ode

Solving a first-order ordinary differential equation using the Runge-Kutta method.

  Implicit Euler Method

euler, ode

Solving a first-order ordinary differential equation using the implicit Euler method (backward Euler method).

  Lennard-Jones Potential

explicit euler method, lennard-jones potential

A simple physical model that approximates the interaction between a pair of neutral atoms or molecules.