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CollocationGrids.jl

A Julia package for numerical computation using collocation grids and spectral methods.

Overview

CollocationGrids.jl provides a comprehensive framework for working with various types of collocation grids commonly used in numerical analysis and spectral methods. The package supports multiple grid types, derivative calculations, interpolation, and multi-dimensional operations.

Features

  • Multiple Grid Types: Support for Chebyshev, Fourier, semi-infinite, and custom grids
  • Multi-dimensional Grids: Tensor product grids for multi-dimensional problems
  • Spectral Differentiation: Accurate computation of derivatives using spectral methods
  • Interpolation: Cardinal function-based interpolation
  • Grid Transformations: Linear rescaling and coordinate transformations

Installation

using Pkg
Pkg.add(url = "https://github.com/inokawazu/CollocationGrids")

Quick Start

using CollocationGrids

# Create a Chebyshev-Lobatto grid with 10 points
grid = ChebyshevLobattoGrid{Float64}(10)

# Get grid points
points = collect(gridpoints(grid))

# Compute first derivative matrix
D1 = derivative_matrix(grid, 1)

# Interpolate a function
f(x) = sin(x)
coefs = f.(points)
interpolated_value = interpolate(grid, coefs, 0.5)

Grid Types

Single-Dimension Grids

  • ChebyshevLobattoGrid{T}(N): Chebyshev-Lobatto grid with N+1 points including boundaries
  • ChebyshevInteriorGrid{T}(N): Chebyshev grid with N interior points (excludes boundaries)
  • FourierGrid{T}(N): Fourier grid with 2N points for periodic problems
  • SemiInfiniteGrid: Grid for semi-infinite domains
  • LinearRescaleGrid: Linearly rescaled version of any grid

Multi-Dimensional Grids

# Create a 2D grid (tensor product)
grid1 = ChebyshevLobattoGrid{Float64}(10)
grid2 = FourierGrid{Float64}(8)
multigrid = MultiGrid(grid1, grid2)

# Get 2D derivative matrices
Dx = derivative_matrix(multigrid, (1, 0))  # ∂/∂x
Dy = derivative_matrix(multigrid, (0, 1))  # ∂/∂y
Dxy = derivative_matrix(multigrid, (1, 1)) # ∂²/∂x∂y

Core Functions

Grid Operations

# Get grid points
points = gridpoints(grid)

# Get domain boundaries
bounds = boundaries(grid)

# Get specific grid point
point_i = gridpoint(grid, i)

Differentiation

# Derivative matrix for order-k derivatives
Dk = derivative_matrix(grid, k)

# Specific derivative element
d_ij = derivative(grid, i, j, order)

# For multi-dimensional grids
derivative_matrices = derivative_array_matrix(multigrid, max_order)

Interpolation

# Cardinal function at point x for basis function j
card_val = cardinal(grid, j, x)

# Interpolate using coefficients
result = interpolate(grid, coefficients, x)

Grid Construction Examples

Chebyshev Grids

# Lobatto grid (includes boundaries at ±1)
clob = ChebyshevLobattoGrid{Float64}(16)

# Interior grid (excludes boundaries)
cint = ChebyshevInteriorGrid{Float64}(16)

Fourier Grids

# For periodic functions on [0, 2π]
fourier = FourierGrid{Float64}(32)

Custom Domain Grids

# Rescale Chebyshev grid to [0, 5]
base_grid = ChebyshevLobattoGrid{Float64}(20)
custom_grid = LinearRescaleGrid(base_grid, 0.0, 5.0)

# Or use the convenience function
rescaler = linear_grid_rescale(0.0, 5.0)
custom_grid = rescaler(base_grid)

Semi-Infinite Domains

# Grid for problems on [0, ∞)
base = ChebyshevLobattoGrid{Float64}(25)
semi = SemiInfiniteGrid(base, 1.0)  # L = 1.0 is the mapping parameter

Advanced Usage

Coordinate Operators

# Get coordinate operators for a grid
X = coordinate_operators(grid)           # Returns diagonal matrix
x_vec = coordinate_vectors(grid)         # Returns vector

# For multi-dimensional grids
X, Y = coordinate_operators(multigrid)   # Tuple of operators

Function Operators

# Create operator from function
f(x) = x^2 + sin(x)
F_op = function_operator(f, grid)        # Diagonal matrix
f_vec = function_vector(f, grid)         # Vector evaluation

Multi-Grid Derivative Arrays

# Get all derivative matrices up to given order
max_order = 3
all_derivatives = derivative_array_matrix(multigrid, max_order)

Type Parameters

All grids are parameterized by element type T (typically Float64 or BigFloat):

# High precision computation
grid_hp = ChebyshevLobattoGrid{BigFloat}(50)

# Standard precision
grid_std = ChebyshevLobattoGrid{Float64}(50)