Finite Differences

📐 Definition

For grid spacing $h>0$ and scalar field $f$,

$$\delta_h^{+} f(x) = \frac{f(x+h)-f(x)}{h},\qquad \delta_h^{-} f(x) = \frac{f(x)-f(x-h)}{h},\qquad \delta_h^{0} f(x) = \frac{f(x+h)-f(x-h)}{2h}$$

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Domain and Codomain

Finite differences apply to functions sampled on uniform grids in $\mathbb{R}$ (and componentwise on $\mathbb{R}^d$). Outputs lie in the same value space as $f$ (scalar or vector valued).

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⚙️ Key Properties

Accuracy (Smooth $f$)

Forward and backward differences are first-order accurate approximations of $f'(x)$; the central difference is second-order accurate.

Linearity and Stencils

Finite differences are linear discrete operators and can be represented as convolution stencils on a uniform grid.

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🎯 Special Cases and Limits

As $h \to 0$, all three differences converge to $f'(x)$ for differentiable $f$. Higher-order stencils extend the same idea using wider supports to raise the order of accuracy.

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🔗 Related Functions

Gradient operators assemble directional differences; fractional derivatives generalize differentiation to non-integer order; convolution operators implement difference stencils via discrete kernels.

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Usage in Oakfield

Oakfield implements finite-difference spatial derivatives as an operator:

• spatial_derivative operator applies finite-difference stencils to real or complex fields (complex is component-wise), with configuration for spacing/method and optional accumulation.
• Boundary behavior is controlled via policy options (periodic vs clamped-style handling) depending on operator configuration and context overrides.

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Historical Foundations

📜 Discretizing Derivatives

Finite differences are among the earliest systematic numerical tools for approximating derivatives, forming a core bridge between continuous PDEs and discrete computation.

🌍 Modern Perspective

They remain standard for grid-based PDE discretizations due to simplicity, locality, and compatibility with conservative flux formulations.

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📚 References

• LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations
• Strikwerda, Finite Difference Schemes and Partial Differential Equations