Module finite_difference 

Finite difference schemes for PDEs in 1D, 2D, and 3D.

Covers explicit/implicit time integration, Crank-Nicolson diffusion, upwind advection, 5th-order WENO, compact finite differences, boundary conditions (Dirichlet/Neumann/periodic), Richardson extrapolation, method of lines, stability analysis (CFL, von Neumann), staggered grids, fractional step method, and ADI (alternating direction implicit).

Structs

FractionalStepResult

Fractional step result from one projection step.

StaggeredGrid1D

A staggered 1D grid holding cell-centred values u and face-centred fluxes f.

Enums

BoundaryCondition

Boundary condition specification for finite difference solvers.

Functions

adi_diffusion_step_2d

One full ADI time step for 2D diffusion ∂u/∂t = α(∂²u/∂x² + ∂²u/∂y²).

advection_beam_warming_step

Beam-Warming second-order upwind advection step (for c > 0).

advection_lax_wendroff_step

Lax-Wendroff second-order advection step for ∂u/∂t + c ∂u/∂x = 0.

advection_periodic_step

First-order upwind advection step on a periodic domain.

advection_upwind_step

First-order upwind advection: advances u one time step dt.

advection_weno5_step

One time step of the WENO-5 advection scheme for ∂u/∂t + c ∂u/∂x = 0.

apply_bc_1d

Applies 1D boundary conditions (left and right) to a solution vector.

apply_dirichlet_2d

Applies 2D Dirichlet boundary conditions to all four edges.

apply_neumann_reflection

Applies Neumann (zero-flux) conditions by ghost-cell reflection.

cfl_number

CFL (Courant-Friedrichs-Lewy) number for the advection equation.

compact_first_derivative

Compact (Padé) 4th-order first derivative.

compact_second_derivative

Compact (Padé) 4th-order second derivative.

diffusion_2d_explicit_step

One explicit 2D diffusion step using the FTCS scheme.

diffusion_crank_nicolson_step

Crank-Nicolson implicit diffusion step.

diffusion_explicit_step

Forward-time central-space (FTCS) explicit diffusion step.

diffusion_implicit_step

Fully implicit (backward Euler) diffusion step.

first_derivative_central

Computes the first derivative of u using second-order central differences.

first_derivative_fourth_order

Computes the fourth-order central difference first derivative.

fractional_step_1d

Fractional step (Chorin projection) method for 1D incompressible flow.

ftcs_amplification

Computes the von Neumann amplification factor magnitude for FTCS diffusion.

ftcs_max_amplification

Maximum amplification factor for the FTCS diffusion scheme over all wave numbers.

is_advection_stable

Returns true if the explicit advection scheme is stable (CFL ≤ 1).

is_diffusion_stable

Returns true if the explicit diffusion scheme (FTCS) is stable.

l1_norm

Computes the discrete L1 norm ||u||₁ = dx * Σ |u[i]|.

l2_error

Computes the discrete L2 error between computed and exact solutions.

l2_norm

Computes the discrete L2 norm ||u||₂ = sqrt(dx * Σ u[i]²).

laplacian_2d

Computes the 2D Laplacian using the standard 5-point stencil.

laplacian_3d

Computes the 3D Laplacian using the 7-point stencil.

linf_norm

Computes the L∞ (maximum) norm of u.

max_dt_advection

Computes the maximum stable time step for explicit advection.

max_dt_diffusion

Computes the maximum stable time step for explicit diffusion (FTCS).

mol_advection_rhs

Method-of-lines spatial operator for the 1D advection equation.

mol_diffusion_rhs

Method-of-lines spatial operator for the 1D diffusion equation.

richardson_derivative

Applies Richardson extrapolation to refine a 1D finite-difference derivative.

richardson_extrapolation

Richardson extrapolation to combine two estimates with step sizes h and h/2.

rk4_step

Runge-Kutta 4 integrator for method of lines.

second_derivative

Computes the second derivative of u with the standard 3-point Laplacian stencil.

thomas_algorithm

Solves the tridiagonal system A x = d using the Thomas algorithm.

total_variation

Computes the total variation TV(u) = Σ |u[i+1] - u[i]|.

uniform_grid_1d

Generates a uniform 1D grid from x_start to x_end with n points.

uniform_grid_2d

Generates a uniform 2D grid (row-major).

von_neumann_diffusion_number

Von Neumann diffusion stability number r = α dt / dx².

wave_equation_step

One explicit time step for the 1D wave equation ∂²u/∂t² = c² ∂²u/∂x².

weno5_flux

Computes the WENO-5 reconstructed flux at interface i + 1/2 for c > 0.

weno5_min_smoothness

Minimum smoothness indicator across stencils (for diagnostic use).