PDEngine.jl

PDEngine.jl is a Julia library designed for solving partial differential equations (PDEs) using a variety of numerical methods. This library aims to provide efficient, flexible, and easy-to-use solvers for some of the most common PDEs encountered in scientific and engineering problems. Documentation

Features

• Heat Equation: Models heat distribution over time.
• Wave Equation: Simulates the propagation of waves through a medium.
• Navier-Stokes Equations: Governs the motion of viscous fluid substances.
• Poisson's Equation: Used in electrostatics, mechanical engineering, and theoretical physics.

Numerical Methods

• Finite Difference Method (FDM): A straightforward approach to discretize PDEs on regular grids.
• Finite Element Method (FEM): Ideal for complex geometries and adaptive mesh refinement.
• Spectral Methods: Provides high accuracy for smooth solutions and periodic boundary conditions.

Usage (Already adjusted for not released v0.0.95, use Manual for accurate usage)

Using the Package:

    using PDEngine

    # Example: Solving the heat equation in 1D
    Δx = 0.01
    Δt = 0.0001
    T = 0.1
    α = 0.01
    N = 100
    temperature_distribution = heat(N, α, T, Δx, Δt) # solving the heat equation with the default (spectral method)

    # Example: Solving Poisson's equation in 2D
    f = zeros(N+1, N+1)  # Example source term
    Δx = 0.01
    poisson_solution = poisson_fem(N, f, Δx)  # solving the poisson equations,set to use with the finite elements method

Currently working on:

v0.1.0 (Full release)

• Parallel computing to get the full speed of julia
• more methods and or PEDs
• ensure all Methods work without any flaws
• ...

Papers used:

• Lehman : Numerical Solutions for PDEs using Mathematica
• Spectral Methods : Spectral Methods and tests by Phillip Schlatter
• ByJus : Representation of a PDE
• ScholarPedia : Introduction to PDEs