BDF, Generalized Alpha, and Runge-Kutta Methods

Problem Description

What numerical methods are available when using a Time-Dependent Solver? What are their relative benefits and when should I use them?

Solution

The Time-Dependent Solver offers three different time stepping methods: The implicit BDF and Generalized alpha methods and the explicit Runge-Kutta family of methods.

The Backward Differentiation Formula (BDF) solver is an implicit solver that uses backward differentiation formulas with order of accuracy varying from one (also know as the backward Euler method) to five. BDF methods have been used for a long time and they 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 may thus get a very smooth solution in time due to the damping in backward Euler. The BDF implementation in COMSOL is of variable order. That is, a high order will be used when possible, and a lower order will automatically be employed when necessary to obtain stability. The BDF method is also a Differential-Algebraic system of Equations (DAE) solver.

Generalized alpha has properties similar to second-order BDF, but the underlying technology is different. It contains a parameter, called alpha in the literature, to control the degree of damping of high frequencies. Compared to BDF (with maximum order two), Generalized alpha has much less damping and is thereby more accurate. For the same reason, it is also less stable. The implementation of Generalized alpha in COMSOL detects which variables are first order in time and which variables are second order in time and applies the correct formulas to the variables. Generalized alpha is not formally a DAE solver even though it frequently is used to solve DAE's anyhow. Generalized alpha was first suggested in 1993 for second order systems of structural mechanics, meaning systems with second-order derivatives in time (Ref. 1). The area of structural mechanics is where the method has been most widely used. In 2000, a formulation for first order systems (the Navier-Stokes equations) was suggested (Ref. 2).

BDF is both more stable and versatile than Generalized alpha but at the same time introduces more damping. Generalized alpha is therefore the default transient solver for applications which are sensitive to excessive dissipation, such as Structural Mechanics, Acoustics, and Transient Electromagnetic Waves. In other applications, such as transport applications, extra robustness is often required and BDF is therefore the default.

The explicit Runge-Kutta family of methods are most appropriate for systems of Ordinary Differential Equations and are usually not as efficient for problems involving Partial Differential Equations.

All of these methods offer the choice between Free, Intermediate, Strict, and Manual timestepping, with Free usually being the default. Free timestepping allows the solver to take larger or smaller timesteps as required to satisfy the specified tolerances. The solvers will try to take as large a timestep as possible, but will reduce the timestep size when necessary if the solution starts to vary rapidly in time. The Intermediate option will force the solver to take at least one timestep within each interval in the Times section of the Time Dependent Study Settings. The Strict option will, in addition, force the solver to take timesteps at the times specified in the Times section. Manual timestepping allows you to specify the timestep taken by the solver which is useful for wave-type problems where you know ahead of time how quickly the solution will change over time. For directions on controlling the timesteps in general, see: Knowledge Base 1254: Controlling the Time Dependent solver timesteps. For directions on using manual timestepping for wave-type problems, see Knowledge Base 1118: Resolving time-dependent waves.

References

1. J. Chung, G.M. Hulbert, *A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-alpha method, *. Appl. Mech. 60 (1993), 371-375.
2. K.E. Jansen, C.H. Whiting, G.M. Hulbert, A generalized-alpha method for integrating the filtered Navier-Stokes equations with a stabilized finite element method, Comput. Methods Appl. Engrg. 190 (2000) 305-319.

See Also

For more information, you can search the Help (press F1 in COMSOL Multiphysics), or open the electronic Documentation (press Ctrl F1 in COMSOL Multiphysics) and then browse to the COMSOL Multiphysics Reference Manual > Studies and Solvers > Solution Operation Nodes and Solvers > About the Time-Dependent Solver section.

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