PDEngine Library Reference

This document provides detailed API documentation for the functions in the PDEngine Library. Each function is described with its signature, arguments, and return values.

Functions

heat_eq_1d_fdm

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

heat_eq_1d_fem

Signature:

function heat_eq_1d_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

heat_eq_1d_crank_nicolson

Signature:

function heat_eq_1d_crank_nicolson(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

navier_stokes_2d_solver

Signature:

function navier_stokes_2d_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

poisson_2d_fdm

Signature:

function poisson_2d_fdm(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

poisson_2d_fem

Signature:

function poisson_2d_fem(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

wave_eq_1d_fdm

Signature:

function wave_eq_1d_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

wave_eq_1d_fem

Signature:

function wave_eq_1d_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