Stability Analysis

PDEs · Axiom Academy

Understanding why numerical solutions stay bounded or blow up A numerical method is stable if errors remain bounded as the computation progresses. If errors grow without bound, the method is unstable and will produce meaningless results. The animation above shows two scenarios: a stable method where errors stay small (green), and an unstable method where errors explode (red). 2. Von Neumann Stability Analysis The Von Neumann method analyzes stability by examining how Fourier modes evolve. We substitute a trial solution of the form: where k is the wave number and g is the amplification factor . This represents how a single Fourier mode grows (or decays) from one time step to the next. The animation shows how different Fourier modes (different wavelengths) are amplified or damped by the numerical scheme. For stability, the amplification factor must satisfy: This ensures that errors do not grow from one time step to the next. If |g| > 1 for any wave number k , the method is unstable. 4. Stability of FTCS for Heat Equation For the Forward Time, Centered Space (FTCS) scheme applied to the heat equation, Von Neumann analysis reveals that stability requires: where r = kΔt/Δx² is the mesh ratio . This is a very restrictive condition: if we halve the spatial step Δx, we must reduce Δt by a factor of 4! 5. Why Implicit Methods are Unconditionally Stable

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