Search: method

27 results

Mechanics 2

  Stabilising an Inverted Pendulum on a Cart

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

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.

  Projectile motion

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.

Astrophysics 1

  Gravity Assist

newton, animation, semi-implicit euler method, gravity

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

Mechanics 1

  Simple Pendulum

animation, ode, explicit euler method

Simulates the simple pendulum and damped simple pendulum

Partial Differential Equations 1

  Relaxation Methods for Solving PDE's

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.

Root Finding 1

  Fixed-Point Iteration

Newton-Rhapson, method

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

Fluid Mechanics 1

  Euler Equations for Inviscid Flow

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

Quantum Mechanics 3

  Numerical Determination of Eigenenergies for an Asymmetric Potential

eigenenergy, forward shooting, eigenstate, schrödinger equation

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

  Band Structures and Newton's Method

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

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

  Numerical Determination of Eigenenergies for the Harmonic Oscillator

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.

Thermodynamics 1

  Martian Atmosphere

temperature, pressure, simpson's method

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

Waves and Acoustics 1


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.

Astrophysics 2

  Planetary Motion - Three Body Problem

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

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

  Planetary Motion

newton, space, explicit euler method, gravity

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

Python Packages 1

  Calling Fortran(95) Routines from a Python Script

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.

Partial Differential Equations 2

  Partial Differential Equations - Two Examples

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

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

  Iterative Gauss-Seidel Method

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

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

Ordinary Differential Equations 4

  Adaptive Runge-Kutta Methods

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

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

  Runge-Kutta Methods

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

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).

  Euler's Method

ode, explicit euler method

Solving a first-order ordinary differential equation with Euler's method.

Root Finding 1

  Newton-Raphson Method


Determining a root with the Newton-Raphson algorithm.

Numerical Integration 2

  Monte Carlo Integration in D Dimensions

monte, carlo

Numerical integration in D dimensions using the Monte Carlo method.

  Monte Carlo Integration in One Dimension

monte, carlo

Numerical integration in one dimension using the Monte Carlo method.

Chemistry 1

  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.

Root Finding 1

  Bisection Method

Determining a root using the bisection method.

Numerical Integration 1

  Numerical Integration

simpson's, method

Numerical integration using the trapezoidal and Simpson's rules.