Search: method

30 results

Fluid Mechanics 1

  Linear Wave Propagation

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.

Ordinary Differential Equations 1

  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.

Basics 1

  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

Electromagnetism 1

  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.

Mechanics 2

  Stabilising an Inverted Pendulum on a Cart

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.

  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.

Astrophysics 1

  Gravity Assist

animation, gravity, newton, semi-implicit euler method

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

Mechanics 1

  Simple Pendulum

animation, explicit euler method, ode

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

method, Newton-Rhapson

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

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.

  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

simpson's method, temperature, pressure

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

Waves and Acoustics 1

  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.

Astrophysics 2

  Planetary Motion - Three Body 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.

  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.

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

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.

  Iterative Gauss-Seidel Method

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.

Ordinary Differential Equations 3

  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).

Root Finding 1

  Newton-Raphson Method

newton

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

method, simpson's

Numerical integration using the trapezoidal and Simpson's rules.