Explicit Methods

PDEs · Axiom Academy

Understanding the FTCS scheme and explicit time-stepping for solving PDEs 1. The FTCS Scheme for the Heat Equation The heat equation describes how temperature diffuses over time: To discretize this equation, we use a forward difference in time and a central difference in space . This gives us the FTCS (Forward-Time Central-Space) scheme. 2. The Explicit Time-Stepping Formula The FTCS discretization leads to an explicit formula for computing the solution at the next time level: where r is the mesh ratio (also called the Courant number for this problem): Explicit methods offer significant computational benefits: Here's a typical implementation in pseudocode: 4. The Critical Limitation: Stability Despite their simplicity, explicit methods have a major drawback: strict stability restrictions . This means that if you refine the spatial grid (make Δx smaller), you must make Δt even smaller (proportional to (Δx)²) to maintain stability. The stability restriction we've seen is a specific example of a more general principle called the Courant-Friedrichs-Lewy (CFL) condition . For the heat equation, the CFL condition manifests as r ≤ 1/2. For other PDEs (like wave equations or advection equations), the CFL condition takes different forms, but the principle remains the same.

This is the written version of the interactive lesson above. See the full PDEs course.