Backward Euler and BDF Methods
Backward differentiation formulas are stiffly stable implicit multistep integrators built for problems with fast, uninteresting decay modes.
Strong damping for stiff systems
Backward Euler is the simplest implicit method: u_{n+1} = u_n + dt f(u_{n+1}). It is only first-order accurate but is both A-stable and L-stable, meaning it damps arbitrarily fast decaying modes strongly. That heavy damping is exactly what stiff problems need, since it lets the step size follow the slow physics without the fast modes destabilizing or polluting the solution.
Backward differentiation formulas (BDF) extend this idea to higher order by fitting a polynomial through several past solution values and requiring its derivative at the new point to match f(u_{n+1}). BDF2 through BDF5 raise accuracy while retaining strong stability for stiff systems.
Stability and order limits
BDF methods are stiffly stable up to order 6; beyond that they lose stability (a consequence of the second Dahlquist barrier), so practical codes use orders one through five. Higher-order BDF methods have a shrinking region of stability, so variable-order implementations lower the order automatically when stability demands it.
Variable step and order
Production stiff solvers implement BDF with adaptive step size and order selection, estimating local error at each step and adjusting both to hold accuracy at minimum cost. Each step still requires a nonlinear solve, typically Newton-based, so a good preconditioner and Jacobian strategy are essential.
- Backward Euler: first-order, A- and L-stable
- BDF fits past states to reach higher order
- Stiffly stable only up to order six
- Variable step and order in production stiff solvers
BDF integrators are the backbone of stiff time-dependent transport and reaction-network solvers, where fast equilibration must be stepped over to reach the slow evolution of interest.