Finite Differences for PDEs

PDEs · Axiom Academy

LESSON Finite Differences for PDEs Master the fundamental techniques for discretizing partial differential equations on computational grids 1. First Derivative Approximations The derivative definition leads to three fundamental difference formulas. Each uses nearby grid points to approximate the continuous derivative at point x i . 2. Second Derivative Approximation The second derivative measures curvature. We build it by taking differences of differences, creating a symmetric three-point stencil that's fundamental to solving PDEs like the heat and wave equations. For 2D PDEs, we need mixed derivatives like ∂²u/∂x∂y. These require a 2D grid stencil that combines differences in both directions. 4. Truncation Error and Accuracy Finite differences are approximations. Taylor series reveals how the error depends on grid spacing Δx, giving us the "order of accuracy." Expand u(x i+1 ) in Taylor series about x i : Rearranging for the forward difference: 5. From PDEs to Difference Equations Replace each derivative in a PDE with its finite difference approximation. The result is an algebraic equation relating values at grid points. Finite difference form (forward time, centered space): 6. Grid Notation and Stencil Visualization Standard notation and visual stencils help communicate finite difference schemes efficiently. Understanding these conventions is essential for reading numerical PDE literature.

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