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Study Notes 2024 Draft views

Finite Difference Methods for Parabolic PDEs

Derivation and analysis of Crank–Nicolson, θ-methods, and higher-order schemes for parabolic equations.

FDM Stability Crank–Nicolson

Overview

Finite difference methods (FDMs) discretize PDEs by replacing derivatives with difference quotients on a mesh. For parabolic equations of the form ut = Lu, we derive and analyze several time-stepping schemes.

The θ-Method Family

The general θ-method for ut = Lu reads:

(un+1 − un) / Δt = θ L un+1 + (1−θ) L un

Special cases: θ = 0 gives explicit Euler (conditionally stable), θ = 1 gives implicit Euler (unconditionally stable, first-order), θ = ½ gives Crank–Nicolson (unconditionally stable, second-order).

Stability Analysis

Von Neumann stability analysis examines the amplification of Fourier modes. For the heat equation ut = αuxx with mesh ratio r = αΔt/Δx²:

  • Explicit Euler: stable iff r ≤ ½
  • Implicit Euler: unconditionally stable
  • Crank–Nicolson: unconditionally stable, amplification factor |g| ≤ 1

Higher-Order Schemes

Fourth-order compact schemes and exponential integrators are surveyed, with attention to their role in option pricing where smooth initial data enables high-order convergence.

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