Time-Dependent Solver

) to find the solution to time-dependent problems (also called dynamic or unsteady problems) using the implicit time-stepping methods BDF or generalized-α or an explicit method from a family of Runge-Kutta methods for solving ordinary differential equations. This solver is automatically used when a Time Dependent study is added to the model.

Also see About the Time-Dependent Solver for information about The Implicit Time-Dependent Solver Algorithms and BDF vs. Generalized-α and Runge–Kutta Methods.

General

Use the list to specify if the settings are synchronized with the corresponding study step (the default). If you select (to override the settings defined in the corresponding study node), you can specify the , , and settings.

Use the list to choose a time unit that is suitable for the time span of the simulation. The default time unit is inherited from the corresponding setting in the study step.

Use the field to enter a vector of times that define the time span for the simulation using the button (

) if needed (default: range(0,0.1,1)).

From the list, choose one of the following options:

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Choose (the default) to store the output times using interpolation from the solver’s time steps.

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Choose to use the time steps from the solver that are closest to a given output time among all the steps that have passed this output time.

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Choose to store the actual time steps from the solver. Specify to store every Nth step in the field (default value: 1; that is, all time steps are stored).

Use the field to enter a positive scalar number (default: 0.01). The solver uses this number to control the relative error in each time step. The tolerance may need to be set to a smaller (tighter) value if the simulation results seem unexpected or inconsistent.

Absolute Tolerance

See Absolute Tolerance Settings for the Time-Dependent Solver for details about this section.

Specify an absolute tolerance that is used by the solver to control the absolute error. The tolerance specified here is applied to all variables unless modified per variable by selecting a method other than the global method for a variable.

Select a to select how the specified absolute tolerance is to be interpreted for the variables that use the global method (by default, all variables use the global method). Select:

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to let the absolute tolerance be applied to scaled variables.

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to let the absolute tolerance be applied to unscaled variables.

Select a to specify how to enter and compute the absolute tolerance. Select:

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to specify a factor (default: 0.1) in the field that makes the absolute tolerance proportional to the relative tolerance. With this method, the absolute error does not become dominating when you reduce the relative tolerance so that it is much smaller than the absolute tolerance.

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to specify a value for the absolute tolerance. In the field, enter a positive number that is applied to either scaled or unscaled variables.

Select a to specify how to enter and compute the absolute tolerance for the time derivative. It can be useful in some cases where control of the absolute tolerances for the time derivatives must be more elaborate in order to reflect the character of the model solved. Select:

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to use an automatic derivative tolerance. The automatic method sets the time derivative factor (see below) to one over the time span.

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to specify a factor (default: 1 Hz) in the field. The time derivative factor acts between the tolerance on the variables and their time derivatives.

Select the check box as needed. See Absolute Tolerance Settings for the Time-Dependent Solver for details.

To specify the absolute tolerance individually for a variable, select from the list and modify the corresponding tolerance using the list. Select:

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to apply the specified tolerance to scaled variables.

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to apply the specified tolerance to unscaled variables.

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(the default) to apply the tolerance specified for the global tolerance.

If or is selected as the , you can specify the absolute tolerance for the selected variable:

Select a to specify how to enter and compute the absolute tolerance. Select:

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to specify values for the absolute tolerances. In the field, enter a positive number that is applied to either scaled or unscaled variables. If a problem of wave-equation type is being solved, then by default, the solver chooses a tolerance for the variable. To manually enter a tolerance for a variable’s time derivatives, use the field.

Select a to specify how to enter and compute the absolute tolerance for the time derivative. Select:

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to use an automatic derivative tolerance. The automatic method sets the time derivative factor to one over the time span.

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to specify a factor (default: 1 Hz) in the field. The time derivative factor acts between the tolerance on the variable and its time derivative.

Time Stepping (General Settings)

Select a time-stepping . See BDF vs. Generalized-α and Runge–Kutta Methods for details about the implicit methods. Select:

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to use a backward differentiation formula.

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to use an explicit method from the Runge-Kutta family of methods for ODEs. From the Runge-Kutta method choose one of the following time-stepping methods:

The methods (the default) combines adaptivity with good stability properties along the imaginary axis. It is therefore suitable for oscillatory problems.

The method is similar to Dormand-Prince 5 (see below) but has an even larger stability region along the negative real axis. It is therefore more efficient for naturally damped problems.

The (DOPRI5) method (see Ref. 13) provides estimates of the accuracy and stability by combining the Runge-Kutta steps using different sets of coefficients to get different order of accuracies.

The Runge-Kutta methods are suitable for nonstiff problems. They do not support DAEs, and it is also not possible to combine them with events or with sensitivity analysis.

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to use the generalized-α method.

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to compute consistent initial values only and then stop. If this option is selected, no other settings are required.

Time Stepping (BDF and Generalized Alpha)

The following settings are available when or is selected above.

If a Fully Coupled or Segregated attribute node is attached to a node, the settings for the nonlinear systems solved by the time-stepping methods come from that node.


	The time-stepping method Generalized alpha requires a Fully Coupled or Segregated attribute node. The time-stepping method BDF can be used without a Fully Coupled or Segregated attribute node. In such a situation, the BDF method uses an internal automatic nonlinear solver.

Steps Taken by Solver

To modify how the time-stepping methods select the time steps, choose an option from the list. Select:

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to let the time-stepping method choose time steps freely. The times specified in the field in the section are not considered when a time step is chosen.

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to force the time-stepping method to take at least one step in each subinterval of the times specified in the field in the section.

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to force the time-stepping method to take steps that end at the times specified in the field in the section. The solver takes additional steps in between these times if necessary.

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to override the automatic choice of time step with a manual choice. Manual time stepping can be useful in cases where the automatic time-step selection does not work; for example, in contact problems, rotating machinery, or fluid-structure interaction.


	Manual is available for BDF and Generalized alpha and overrides the local error estimation made in each time step.

For all settings for except , the check box is selected by default. The solver can then step past the last time in the list, leading to the end time being interpolated. If you clear this check box, the solver will not step past the last time in the list, so that the time stepping includes the last time and takes no steps past that time.

Further options that apply to one or several combinations (as indicated at each bullet) of selections made in the and lists are:

If , , or is selected for the :

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. By default, the solver chooses an initial step automatically. Select the check box for manual specification of an initial step. By default, the first step is 0.1% of the end time, which can affect your solution so that you do not get the same time-step history up to a certain earlier time when the end time is changed. This should, however, not change the result appreciably if you use tight enough tolerances for that the automatic time step control. Also, the results of the consistent initialization can be strongly dependent on the initial time step taken, if the initial conditions and the boundary conditions do not match, for example. If needed, you can change the values for the or the (see below).

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. By default, the solver chooses a maximum time step automatically. Select as the maximum step constraint for manual specification of a fixed maximum time step. A constant maximum step constraint is a positive scalar value, which can be an expression that evaluates to a numerical value before entering the solver. The expression can include global parameters. Select as the maximum step constraint for more general expressions of the allowed maximum time step. These expressions are evaluated while solving and can, for instance, depend on the time parameter itself.


	The solver uses the absolute value of the expression for the maximum step constraint.

Additionally, if is selected as the time stepping :

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(available if , , or is selected for the ). This setting controls the maximum allowed degree of the interpolating polynomial of the BDF method.

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(available if , , or is selected for the ). This setting can be used to prevent the solver from decreasing the order of the BDF method below 2.

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(available if is selected for the ). The order of the BDF can be 1–5 (default order: 2).

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(available if is selected for the ). During the startup of the BDF method, a shorter time step will be used to compensate for the lower order that is used for the first handful of steps. The initial step is a fraction of the first step, and the solver then exponentially increases the step length until the requested step length is reached. This settings and the initial step growth rate below control that startup phase. The default values depend on the selected BDF order.

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(available if is selected for the ).

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(available if is selected for the ). Enter a manual time step specification as a scalar, a vector of times, or an expression containing global variables or parameters in the field. The relative and absolute tolerances are still used to terminate the algebraic equations at each time step. Also, the requested time step will be reduced if the algebraic solver does not converge.

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. This setting can be used to set the event tolerance (default value: 0.01), which is used for root finding of event conditions when using implicit events; see Explicit Event.


	This does not apply to the start-up phase of the simulation.

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. Select this check box to use a nonlinear controller for more efficient time-step control in the BDF method, especially for highly nonlinear problems such as multiphase flow and turbulence in fluid dynamics. When nonlinear failures occur, the nonlinear controller becomes active and uses a more careful time step control. The nonlinear controller acknowledges that the step size for Newton stability might be smaller than the step size for BDF accuracy.

If is selected as the time stepping :

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(available if , , or is selected for the ). Select this check box and enter a positive integer in the field to make the solver more restrictive when increasing the time step. This integer is the number of time steps taken by the solver before the increase of the time step is actually performed, from the first step where the error estimator signals that the current step is too small. This setting is useful when there is a natural variation in the solution, like periodicity or quasi-periodicity, which make the time steps vary up and down in size. The generalized-α method does not work well when the time step changes often, so in those situations it is better to damp the changes by a more conservative strategy using this setting. Entering 0 results in the same behavior as clearing the check box.

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(available if is selected for the ). Enter a manual time step specification as a scalar, a vector of times, or an expression containing global variables or parameters in the field.

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. Enter a number between 0 and 1 to control how much damping of high frequencies the solver provides. A value close to 0 results in efficient damping, while a number close to 1 results in little damping.

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. Select to use linear extrapolation of the present solution to construct the initial guess for the nonlinear system of equations to be solved at the next time step. Select to use the current solution as initial guess.

Also, the following settings are available for the BDF and generalized-α methods:

Algebraic Variable Settings

Use the list to control whether the solver automatically determines if a system includes a differential-algebraic equation or not. Select:

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to make the solver look for zero-filled rows or columns in the mass matrix as a means of detecting a differential-algebraic equation.

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, if the model includes a differential-algebraic equation where the mass matrix has no zero-filled rows or columns.

Use the list to control how the solver performs consistent initialization of differential-algebraic systems. The consistent initialization procedure tries to reconcile inconsistent initial values. In doing that, it takes a small time step (by default, 0.1% of the initial step size). If, for example, you have boundary conditions and initial conditions that do not match, the results of the consistent initialization is strongly dependent on the initial time step taken. Select:

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(the default) to perform consistent initialization using a small artificial step with the backward Euler method. When this is selected, enter a value in the field. This value is a dimensionless quantity that determines the size of the time step for the backward Euler method (in terms of the initial step). Adjusting this value can improve the accuracy of the initialization step but can also affect the start-up of some models. The default value is 0.001 (that is, the small backward Euler step size is 0.1% of the initial step size).

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to indicate that the initial values already are consistent, which means that the solver does not modify them.

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to use a consistent initialization routine that is preferable to for index-1 differential-algebraic equations.


	The On option is only available when Time method is set to BDF at the same time that the internal nonlinear solver of the BDF method is used.

Use the list to control how to treat algebraic degrees of freedom of a differential-algebraic system when estimating the time discretization error. Select:

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(the default) to include the algebraic degrees of freedom in the error norm.

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to exclude the algebraic degrees of freedom from the error norm.

Excluding algebraic degrees of freedom (which stem from stationary equations in the model) means that the algebraic variables are not included in the error test for the time step. The algebraic variables are still solved for as part of the general system of equations. Excluding the algebraic variables from the error test might have the effect that the constraints (including hidden constraints, which are implicitly part of the equations) are not accurately fulfilled. In general, excluding algebraic degrees of freedom is not recommended when solving DAE systems of index 1, whereas it can be generally encouraged for DAE systems of index 2 (see Ref. 6).

If is selected from the list, select the check box if you want to perform an rescaling of the variables. In fluid-flow simulations, for example, the scaling for velocity can be problematic because the velocity is often zero at the start but then changes dramatically after a consistent initialization. Rescaling of the velocity and pressure can provide a smoother start and avoid using very small time steps.

Time Stepping (Runge-Kutta Methods)

The following settings are available when is selected as the time-stepping method:

Steps Taken by Solver

To modify how the Runge-Kutta time-stepping methods select the time steps, choose an option from the list. Select:

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to let the time-stepping method choose time steps freely. The times specified in the field in the section are not considered when a time step is chosen.

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to force the time-stepping method to take at least one step in each subinterval of the times specified in the field in the section.

•

to force the time-stepping method to take steps that end at the times specified in the field in the section. The solver takes additional steps in between these times if necessary.

•

to override the automatic choice of time step with a manual choice.

Further options that apply to one or several combinations (as indicated at each bullet) of selections made in the and lists are:

If , , or is selected for the :

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. By default the solver chooses an initial step automatically. Select the check box for manual specification of an initial step.

The following settings for , , and are only available for the Dormand-Prince time-stepping method:

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. By default the solver chooses a maximum time step automatically. Select the check box for manual specification of a maximum time step. The maximum time step is a positive scalar value, which can be an expression that evaluates to a numerical value. The expression can include global parameters.

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and . These growth ratio limits restrict how fast the step size may change, enforcing that the values of the ratio hnew/hold is within the minimum step size growth ratio (default: 0.2) and the maximum step size growth ratio (default: 10).

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. The solver multiplies this factor (default: 0.9) to the estimated largest allowed step size to avoid taking too large step sizes when the estimate overshoots.

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. This setting affects the behavior of the PI (proportional-integral) controller that adds damping on step size changes to avoid choosing too large steps, which would then be rejected. The default value is , which corresponds to a PI controller that responds quickly to changes. The option sets the controller to react more slowly, giving smoother choices of time steps. You can also turn off the PI controller by selecting . This setting affects the parameters α and β in the relation

. Here S is the safety factor described above, and erri is the estimated error in step i.

If is selected for the :

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. Enter a manual time step specification as a scalar, a vector of times, or an expression containing global variables or parameters in the field.

For all options in the list, you can also activate or turn off detection of numerical stiffness using the check box (selected by default). If active, the stiffness detection uses a mechanism to detect if the problem that you solve becomes numerically stiff (which means that an explicit time stepping is required to take very small time steps). If the problem is considered to be stiff, an error appears and the solver stops. You can then switch to another solver that is better suited for stiff problems, such as an implicit BDF method.

Results while Solving

This section mirrors what is defined for the Time Dependent node’s section. That is, changes made to the Time-Dependent node are reflected here.

Output

Use the list to control at what times the solver stores a solution. Select:

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to store solutions at the values entered in the field in the section.

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to store solutions at the time steps taken by the solver.


	The selection made in the list Steps taken by solver in the Time Stepping section influences the output in this situation.

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Select the check box to compute and store reaction forces in the output. This option is not supported when using any of the Runge-Kutta time-stepping methods.

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The computation of boundary flux variables involves solving a system of equations to obtain a continuous field from nodal flux values. If the check box is selected, this system of equations is lumped. The benefits of using this option is that it can avoid certain spurious oscillations in the computed flux field and it is also slightly faster. Lumping is not suitable in 3D for shape functions of order higher than 1. Lumping is not supported when using any of the Runge-Kutta time-stepping methods.

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Select the check box to store time derivatives of the variables solved for in the output. Storing the time derivatives gives more accurate results when evaluating quantities that involve these time derivatives.

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Select the check box to store two additional solutions every time an implicit or explicit event is triggered. See The Events Interface. This stores the solutions before and after the reinitialization.

Advanced

Select the check box to be able to solve problems that are not automatically determined to be complex-valued in a correct way.

Least-Squares Data


	The section only appears if your license includes the Optimization Module and you are solving a time-dependent least-squares optimization problem.

Least-squares times are read from files pointed out by the least-squares objective nodes in an Optimization interface. If there is a least-squares objective with a time column, the time values are displayed under . If is on (which is the default), the least-squares times are read and merged with the user-defined time list at runtime. The merged data governs the setup of the time-dependent solver.

General parameter values list here refers to the list of times in the section above. If is on (which is the default), start and end simulation times that you have provided are respected when merging with time values from files. Time values from files outside the user-defined time range are ignored. Otherwise (that is, is off), all time values from files are used and merged with the user-defined time list.

If is off, no time values from files are used.

You can change the default values of and only if is set to .

Constants

In this section you can define constants that can be used as temporary constants in the solver. You can use the constants in the model or to define values for internal solver parameters. These constants overrule any previous definition (for example, from Global Definitions). Some examples of when it can be useful to define constants for a solver:

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When doing more advanced solver sequences, where constants need to be changed between calls (for example, in for-loops).

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When you do not want to change the global definition of a parameter or when you cannot or do not want to use a Parametric Solver feature.

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When you want to define auxiliary parameters that are part of the equations like CFLCMP or niterCMP and where the solver does not define these parameters (niterCMP is defined by the nonlinear solvers).

Click the button (

) to add a constant and then define its name in the column and its value (a numerical value or parameter expression) in the column. By default, any defined parameters are first added as the constant names, but you can change the names to define other constants. Click (

) to remove the selected constant from the list.

Log

This section, which is initially empty, contains a log from the time stepping. This log is stored in the MPH-file. Select the to keep warning messages in this log so that the information in those warnings is available also when reopening the model.


	The Log Window (The Time-Dependent Solver Log)


	The Black-Scholes Equation: Application Library path COMSOL_Multiphysics/Equation_Based/black_scholes_put