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PDEngine Library Manual

Welcome to the manual for the PDEngine Library. This document provides detailed instructions on how to use the functions available in the library for solving various partial differential equations (PDEs) using Julia.

Heat Equation Solver

Finite Difference Method (FDM)

Function Signature: function heat_eq_1d_fdm(N, α, T, Δx, Δt)

Description: Solves the 1D heat equation using the finite difference method.

Arguments:

  • N: Number of grid points (excluding boundaries)
  • α: Thermal diffusivity
  • T: Total simulation time
  • Δx: Spatial step size
  • Δt: Temporal step size

Returns:

  • u: Final temperature distribution

Example Usage: To solve the 1D heat equation using FDM with a grid of 50 points, thermal diffusivity of 1, and a total simulation time of 1.0, use the following parameters:

  • N = 50
  • α = 1
  • T = 1.0
  • Δx = 1.0 / N
  • Δt = 0.001

Call the function as follows:

u_fdm = heat_eq_1d_fdm(N, α, T, Δx, Δt)

Finite Element Method (FEM)

Function Signature: function heateq1d_fem(N, α, T, Δx, Δt)

Description: Solves the 1D heat equation using the finite element method.

Arguments:

N: Number of grid points (excluding boundaries) α: Thermal diffusivity T: Total simulation time Δx: Spatial step size Δt: Temporal step size Returns:

u: Final temperature distribution Example Usage: To solve the 1D heat equation using FEM with the same parameters as above, use:

N = 50 α = 1 T = 1.0 Δx = 1.0 / N Δt = 0.001 Call the function as follows:

u_fem = heat_eq_1d_fem(N, α, T, Δx, Δt)

Crank-Nicolson Method (CN)

Function Signature: function heateq1dcranknicolson(N, α, T, Δx, Δt)

Description: Solves the 1D heat equation using the Crank-Nicolson method.

Arguments:

N: Number of grid points (excluding boundaries) α: Thermal diffusivity T: Total simulation time Δx: Spatial step size Δt: Temporal step size Returns:

u: Final temperature distribution Example Usage: To solve the 1D heat equation using CN with the same parameters, use:

N = 50 α = 1 T = 1.0 Δx = 1.0 / N Δt = 0.001 Call the function as follows:

u_cn = heat_eq_1d_crank_nicolson(N, α, T, Δx, Δt)

2D Navier-Stokes Equations

Function Signature: function navierstokes2d_solver(N, Re, T, Δx, Δt)

Description: Solves the 2D Navier-Stokes equations for incompressible flow.

Arguments:

N: Number of grid points in each dimension (excluding boundaries) Re: Reynolds number T: Total simulation time Δx: Spatial step size Δt: Temporal step size Returns:

u: Velocity field p: Pressure field Example Usage: To solve the 2D Navier-Stokes equations with a grid of 50x50 points and a Reynolds number of 100, use:

N = 50 Re = 100 T = 1.0 Δx = 1.0 / N Δt = 0.001 Call the function as follows:

u, p = navier_stokes_2d_solver(N, Re, T, Δx, Δt)

Poisson's Equation Solver

Finite Difference Method (FDM)

Function Signature: function poisson2dfdm(N, f, Δx)

Description: Solves 2D Poisson's equation using the finite difference method. Poisson's equation is given by: ∇²u = -f where f is a source term.

Arguments:

N: Number of grid points along each dimension (excluding boundaries) f: Source term (function or matrix) Δx: Spatial step size Returns:

u: The final solution field Example Usage: To solve 2D Poisson's equation with a source term and a grid of 50x50 points:

N = 50 f = ... (define your source term) Δx = 1.0 / N Call the function as follows:

u = poisson_2d_fdm(N, f, Δx)

Finite Element Method (FEM)

Function Signature: function poisson2dfem(N, f, Δx)

Description: Solves 2D Poisson's equation using the finite element method (FEM). Poisson's equation is given by: ∇²u = -f where f is a source term.

Arguments:

N: Number of grid points along each dimension (excluding boundaries) f: Source term (function) Δx: Spatial step size Returns:

u: The final solution field in grid format Example Usage: To solve 2D Poisson's equation using FEM with a grid of 50x50 points:

N = 50 f = ... (define your source term) Δx = 1.0 / N Call the function as follows:

u = poisson_2d_fem(N, f, Δx)

Wave Equation Solver

Finite Difference Method (FDM)

Function Signature: function waveeq1d_fdm(N, c, T, Δx, Δt)

Description: Solves the 1D wave equation using the finite difference method. The wave equation is given by: ∂²u/∂t² = c² ∇²u where c is the wave speed.

Arguments:

N: Number of spatial grid points (excluding boundary points) c: Wave speed T: Total simulation time Δx: Spatial step size Δt: Temporal step size Returns:

u: The final displacement field Example Usage: To solve the 1D wave equation with a grid of 50 points and a wave speed of 1.0:

N = 50 c = 1.0 T = 1.0 Δx = 1.0 / N Δt = 0.001 Call the function as follows:

u = wave_eq_1d_fdm(N, c, T, Δx, Δt)

Finite Element Method (FEM)

Function Signature: function waveeq1d_fem(N, c, T, Δx, Δt)

Description: Solves the 1D wave equation using the finite element method (FEM). The wave equation is given by: ∂²u/∂t² = c² ∇²u where c is the wave speed.

Arguments:

N: Number of spatial grid points (excluding boundary points) c: Wave speed T: Total simulation time Δx: Spatial step size Δt: Temporal step size Returns:

u: The final displacement field Example Usage: To solve the 1D wave equation using FEM with a grid of 50 points:

N = 50 c = 1.0 T = 1.0 Δx = 1.0 / N Δt = 0.001 Call the function as follows:

julia u = wave_eq_1d_fem(N, c, T, Δx, Δt)