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

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

embedded runge-kutta pair, integration, set of odes

The trajectory of a charged particle propagating in a non-uniform magnetic field is calculated by solving the Lorentz force law using an embedded Runge-Kutta pair. The results show that the particle is mirrored.

implementation, basic, ode, explicit euler method

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

magnetism, metropolis, equilibrium, autocorrelation, monte carlo, ising, spin

Using the Metropolis algorithm to approximate the magnetization and specific heat for a 2D Ising lattice.

magnetism, Lorentz' law, ode, explicit euler method

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

eigenenergy, eigenstate, schrödinger equation, waves, spin, hemmer

Solving the time independent Schrödinger equation in one dimension using matrix diagonalisation for five different potentials.

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

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

Solving the Friedmann equations to model the expansion of our universe.

ode, gravity, 4th order runge-kutta, einstein, angular momentum, space, fortran, extrapolation, f2py

Computes the precession of Mercury by linear extrapolation.

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

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

animation, ode, explicit euler method

Simulates the simple pendulum and damped simple pendulum

differential equation, ode, gravity, newton, 4th order runge-kutta, einstein, angular momentum, space

Discussion of orbits in the Schwarzschild Geometry.

chebychev nodes, interpolation, curve fitting, system of equations, runge's phenomenon, sparse matrix

Uses cubic splines to interpolate a given set of data points

Solving the one-dimensional stationary heat equation with a Gaussian heat source by approximating the solution as a sum of Lagrange polynomials.

polymer, hemmer, hansen, lennard-jones potential, sparse matrix, nygård, elasticity

Simulates the disentanglement of a polymer from a surface using an increasing electric field

specific heat, partition function, magnetism, spin

Computing the internal energy, specific heat and magnetisation in the 1D and 2D Ising model. An analytical solution to the XY model is also provided.

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

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

chebychev nodes, newton, interpolation, lagrange, runges phenomenon

Using polynomial interpolation to interpolate a set of points and to approximate a function or a curve.

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

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

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

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).

Solving a second-order ordinary differential equation (Newton's second law) using Verlet integration.

explicit euler method, lennard-jones potential

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