Comsol Multiphysics (a popular FEM package) includes two time-stepping algorithms (IDA aka BDF, and Generalized-alpha), described in their documentation as follows (quoted here under Fair Use; emphasis mine):
• IDA was created at the Lawrence Livermore National Laboratory (Ref. 4) and is a modernized implementation of the DAE solver DASPK, which uses variable-order variable-step-size backward differentiation formulas (BDF).
• Generalized-α is an implicit, second-order accurate method with a parameter α or ρ∞ (0 ≤ ρ∞ ≤ 1) to control the damping of high frequencies. With ρ∞ = 1, the method has no numerical damping. For linear problems this corresponds to the midpoint rule. ρ∞ = 0 gives the maximal numerical damping; for linear problems the highest frequency is then annihilated in one step. The method was first developed for second-order equations in structural mechanics and later extended to first-order systems (Ref. 7).
BDF methods have been used for a long time and are known for their stability. However, they can have severe damping effects, especially the lower-order methods. Backward Euler severely damps any high frequencies. Even if you are expecting a solution with sharp gradients, you might get a very smooth solution due to the damping in backward Euler.
Compared to BDF (with maximum order two), generalized-α causes much less damping and is thereby more accurate. For the same reason it is also less stable.
- A.C. Hindmarsh, P.N. Brown, K.E. Grant, S.L. Lee, R. Serban, D.E. Shumaker, and C.S. Woodward, “SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers,” ACM T. Math. Software, vol. 31, p. 363, 2005.
- K.E. Jansen, C.H. Whiting, G.M. Hulbert, “A Generalized-α Method for Integrating the Filtered Navier–Stokes Equations with a Stabilized Finite Element Method,” Comput. Methods Appl. Mech. Engrg., vol. 190, pp. 305–319, 2000.
This makes good sense, but I'd like to understand the essence of this numerical damping phenomenon. Is there a minimal example somewhere (1D is best, if possible) that shows numerical damping clearly, and allows to see where it comes from?