Crank-Nicolson Method

PDEs · Axiom Academy

The Gold Standard for Parabolic PDEs: Averaging Explicit and Implicit Schemes 1. Averaging Explicit and Implicit Schemes The Crank-Nicolson method is defined as the arithmetic average of FTCS (Forward-Time Central-Space) and BTCS (Backward-Time Central-Space): This means we evaluate the spatial derivative at the midpoint in time between levels n and n+1. The resulting scheme is: 2. Second-Order Accurate in Space and Time The Crank-Nicolson method achieves O(Δt²) + O(Δx²) accuracy. This is a significant improvement over FTCS and BTCS, which are only first-order accurate in time. Time discretization: Second-order (centered in time) Space discretization: Second-order (central differences) Overall error: O(Δt²) + O(Δx²) The averaging of explicit and implicit schemes at consecutive time levels creates a centered approximation in time, which is what gives us the second-order temporal accuracy. Like BTCS, the Crank-Nicolson method is unconditionally stable for parabolic PDEs. This means it remains stable for any choice of Δt and Δx, not just when r = αΔt/Δx² ≤ 1/2. The amplification factor G satisfies |G| ≤ 1 for all Fourier modes, regardless of the mesh ratio r = αΔt/Δx². Rearranging the Crank-Nicolson equation leads to a tridiagonal linear system that must be solved at each time step: This can be written in matrix form as Au n+1 = b , where: Right-hand side: depends on u n

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