The following background information about the Time-Dependent Solver discusses these topics: The Implicit Time-Dependent Solver Algorithms and BDF vs. Generalized-α and Explicit Methods. Also see Selecting a Stationary, Time-Dependent, or Eigenvalue Solver.

which is often referred to as the method of lines. Before solving this system, the algorithm eliminates the Lagrange multipliers Λ. If the constraints 0 = M are linear and time independent and if the constraint force Jacobian NF is constant, then the algorithm also eliminates the constraints from the system. Otherwise, it keeps the constraints, leading to a differential-algebraic system.

In COMSOL Multiphysics, the IDA and generalized-a solvers are available to solve the above ODE or DAE system:

For implicit time-stepping schemes, a nonlinear solver is used to update the variables at each time step. The nonlinear solver used is controlled by the active Fully Coupled and Segregated solver subnodes. These subnodes provide much control of the nonlinear solution process: It is possible to choose the nonlinear tolerance, damping factor, how often the Jacobian is updated, and other settings such that the algorithm solves the nonlinear system more efficiently.

For the BDF (IDAS) solver there is another alternative, and that is to use the built-in nonlinear solver IDAS. This solver is used when all Fully Coupled and Segregated nodes are disabled. The linear solver is in this case controlled by the active linear solver subnode.

where K = −∂L/∂U is the stiffness matrix,

is the mass matrix. When E = 0, D is often called the mass matrix.

When using IDA for problems with second-order time derivatives (E ≠ 0), extra variables are internally introduced so that it is possible to form a first-order time-derivative system (this does not happen when using generalized-α because it can integrate second-order equations). The vector of extra variables, here Uv, comes with the extra equation

where U denotes the vector of original variables. This procedure expands the original ODE or DAE system to double its original size, but the linearized system is reduced to the original size with the matrix E + σ D + σ2 K, where σ is a scalar inversely proportional to the time step. By the added equation, the original variable U is therefore always a differential variable (index-0). The error test excludes the variable Uv unless consistent initialization is on, in which case the differential Uv-variables are included in the error test and the error estimation strategy applies to the algebraic Uv-variables.

For the Time-Dependent Solver under the section Absolute Tolerance, the absolute and relative tolerances control the error in each integration step. More specifically, let U be the solution vector corresponding to the solution at a certain time step, and let E be the solver’s estimate of the (local) absolute error in U committed during this time step. For the Unscaled method, the step is accepted if

where Aus,i is the unscaled absolute tolerance for DOF i, R is the relative tolerance, M is the number of fields, and Nj is the number of degrees of freedom in field j. The numbers Aus,i are computed from a conversion of the input value Ak for the corresponding dependent variable k. For degrees of freedom for Lagrange shape functions or for ODEs, these values are the same as entered (that is, Aus,i = Ak), but for vector elements there is a field-to-DOF conversion factor involved.

For the Scaled method and when you select Update scaled absolute tolerance, the step is accepted if

where EY is the solver’s estimate of the (local) absolute error in Y, As,i is the scaled absolute tolerance for DOF i, M is the number of fields, R is the relative tolerance, Nj is the number of degrees of freedom in field j, and Yi is the scaled solution vector. For dependent variables that are using the scaling method Automatic, the numbers As,i are computed from the input values Ak according to the formula

where α = 1/5, j is the time-step iteration number j = 0,1,…, and ,are the 2-norm and maximum norm of the dependent variable ki , respectively. Here is the converted input value Ak for the field k and DOF i. For dependent variables that are using another scaling method or when the Update scaled absolute tolerance check box is cleared, then .

Note that with the Tolerance method set to Factor, the absolute tolerance is specified as a factor of the relative tolerance. If you use a specific absolute tolerance instead, that value can reduce the effect of further reductions of the relative tolerance (so that when , the effect of further reductions of the relative tolerance diminishes).

The generalized-α (alpha) solver has properties similar to the second-order BDF solver but the underlying technology is different. It contains a parameter, called α in the literature, to control the degree of damping of high frequencies. 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.

The implementation of the generalized-α method in COMSOL Multiphysics detects which variables are first order in time and which variables are second order in time and applies the correct formulas to the variables.

In many cases, generalized-α is an accurate method with good enough stability properties, but the BDF method is more robust and better at handling changes to the time step. The generalized-α method is preferred for simulations when the time step is almost constant and when small or minimal damping is needed, such as pure vibration problems. For that reason, some physics interfaces in COMSOL Multiphysics — for structural mechanics and electromagnetic waves, for example — use generalized-α as the default transient solver. Some complicated problems, however, need the extra robustness provided by the BDF method.

The About Auxiliary Equation-Based Nodes section is tailored toward explicit time stepping. The method is quadrature free as well as matrix free. Only element local matrices are formed. A suitable stable time step can be determined automatically by specifying the variable wahw.wtc, which should be an estimate of the maximum wave speed for the equations in the interface. The time-explicit algorithm then translates this speed to a local cell time scale. For a global time marching method like Runge–Kutta or Adams–Bashforth 3, the time step is directly related to the smallest cell time scale. When there is a large difference in cell time scales, a global time marching method is not very efficient. For this reason, there is also a local time-marching method, Adams–Bashforth 3 (local), which divides the cells into groups based on the cell time scale. The groups are then time marched with different time-step sizes, making this a more efficient method.