trapezoidal method, waves, Fourier transform

An introduction to studying linear surface waves on an infinite domain. In particular, the problem of finding the time evolution of a small perturbation of the surface of an inviscid and incompressible fluid.

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.

explicit euler method, ode, implementation, basic

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

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.

animation, 4th order runge-kutta, system of equations

The 4th order Runge-Kutta method was used to integrate the equations of motion for the system, then the pendulum was stabilised on its inverted equilibrium point using a proportional gain controller and linear quadratic regulator.

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.

animation, gravity, newton, semi-implicit euler method

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

poisson's equation, iterative, laplace's equation, uniqueness theorem

The Jacobi, Gauss-Seidel and Successive overrelaxation (SOR) methods are introduced and discussed with the Poisson equation as an example.

Solving fixed-point problems using the Fix-Point Iteration method.

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

eigenenergy, forward shooting, eigenstate, schrödinger equation

Using a forward-shooting method to determine the eigenenergies and eigenfunctions of an asymmetric potential in one dimension.

schrödinger equation, bloch's theorem, newton's theorem

Using Newton's method to calculate the band structure for the simple Dirac comb potential in one dimension.

eigenenergy, harmonic oscillator, forward shooting, eigenstate

Using the method of forward shooting to determine numerically the eigenenergies of the quantum harmonic oscillator in one dimension.

simpson's method, temperature, pressure

Computing planet Mars' atmospheric pressure profile from its temperature profile.

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.

animation, space, gravity, newton, embedded runge-kutta pair, angular momentum

Applying the fourth order Runge-Kutta method and the adaptive step size Runge-Kutta method to calculate the trajectories of three bodies.

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.

electricity, fortran, trapezoidal method

Making use of the Fortran to Python package F2PY which enables creating and compiling a Fortran routine before converting it to a Python Module, which can be imported to any Python script.

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.

sparse matrix, system of equations, iterative, laplace's equation, pde, differential equation, gauss

Solves a linear of system of equations using the iterative Gauss-Seidel method.

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.

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

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

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

Numerical integration in D dimensions using the Monte Carlo method.

Numerical integration in one dimension using the Monte Carlo method.

explicit euler method, lennard-jones potential

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

Numerical integration using the trapezoidal and Simpson's rules.