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Use the Time unit 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.
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Use the Times field to enter a vector of times that define the time span for the simulation using the Range button () if needed (default: range(0,0.1.1)).
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Use the Relative tolerance 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.
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Scaled to apply the specified tolerance to scaled variables.
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Unscaled to apply the specified tolerance to unscaled variables.
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Use global (the default) to apply the tolerance specified for the global tolerance.
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Enter a Tolerance value to modify the absolute tolerance for the selected variable.
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If a problem of wave-equation type is being solved, and if Method in the Time Stepping section is set to BDF, then by default, the solver chooses a tolerance for these components. To manually enter a tolerance for a time derivative when using a first-order time integration method like BDF, select the Tolerance, time derivative check box and enter a tolerance in the associated field. The generalized-α method does not use this tolerance setting.
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The Method setting (Scaled or Unscaled) that is selected for a variable applies also to its time derivative.
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BDF to use a backward differentiation formula.
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Runge-Kutta 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:
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RK34 (the default) combines adaptivity with good stability properties along the imaginary axis. It is therefore suitable for oscillatory problems.
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Cash-Karp 5 is similar to Dormand-Prince 5 but has an even larger stability region along the negative real axis. It is therefore more efficient for naturally damped problems.
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Dormand-Prince 5 (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.
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Generalized alpha to use the generalized-α method.
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Initialization only to compute consistent initial values only and then stop. If this option is selected, no other settings are required.
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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.
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Free to let the time-stepping method choose time steps freely. The times specified in the Times field in the General section are not considered when a time step is chosen.
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Intermediate to force the time-stepping method to take at least one step in each subinterval of the times specified in the Times field in the General section.
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Strict to force the time-stepping method to take steps that end at the times specified in the Times field in the General section. The solver takes additional steps in between these times if necessary.
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Manual 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.
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Manual is available for BDF and Generalized alpha and overrides the local error estimation made in each time step.
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Initial step. By default, the solver chooses an initial step automatically. Select the Initial step 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 Initial step or the Fraction of initial step for Backward Euler (see below).
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Maximum step constraint. By default, the solver chooses a maximum time step automatically. Select Constant 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 Expression 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.
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Maximum BDF order (available if Free, Intermediate, or Strict is selected for the Steps taken by solver). This setting controls the maximum allowed degree of the interpolating polynomial of the BDF method.
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Minimum BDF order (available if Free, Intermediate, or Strict is selected for the Steps taken by solver). This setting can be used to prevent the solver from decreasing the order of the BDF method below 2.
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BDF order (available if Manual is selected for the Steps taken by solver). The order of the BDF can be 1–5 (default order: 2).
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Initial step fraction (available if Manual is selected for the Steps taken by solver). 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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Time step (available if Manual is selected for the Steps taken by solver). Enter a manual time step specification as a scalar, a vector of times, or an expression containing global variables or parameters in the Time step 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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Event tolerance. 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.
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Nonlinear controller. 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.
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Time step increase delay (available if Free, Intermediate, or Strict is selected for the Steps taken by solver). 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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Time step (available if Manual is selected for the Steps taken by solver). Enter a manual time step specification as a scalar, a vector of times, or an expression containing global variables or parameters in the Time step field.
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Amplification for high frequency. 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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Predictor. Select Linear 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 Constant to use the current solution as initial guess.
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Maybe 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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Yes if the model includes a differential-algebraic equation where the mass matrix has no zero-filled rows or columns.
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Backward Euler (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 Fraction of initial step for Backward Euler 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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Off to indicate that the initial values already are consistent, which means that the solver does not modify them.
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On to use a consistent initialization routine that is preferable to Backward Euler for index-1 differential-algebraic equations.
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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.
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Include algebraic (the default) to include the algebraic degrees of freedom in the error norm.
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Exclude algebraic to exclude the algebraic degrees of freedom from the error norm.
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Free to let the time-stepping method choose time steps freely. The times specified in the Times field in the General section are not considered when a time step is chosen.
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Intermediate to force the time-stepping method to take at least one step in each subinterval of the times specified in the Times field in the General section.
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Strict to force the time-stepping method to take steps that end at the times specified in the Times field in the General section. The solver takes additional steps in between these times if necessary.
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Manual to override the automatic choice of time step with a manual choice.
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Initial step. By default the solver chooses an initial step automatically. Select the Initial step check box for manual specification of an initial step.
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Maximum step. By default the solver chooses a maximum time step automatically. Select the Maximum step 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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Minimum step size growth ratio and Maximum step size growth ratio. 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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Step size safety factor. 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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PI step controller. 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 Quick, which corresponds to a PI controller that responds quickly to changes. The Smooth option sets the controller to react more slowly, giving smoother choices of time steps. You can also turn off the PI controller by selecting Disabled. 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.
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Time step. Enter a manual time step specification as a scalar, a vector of times, or an expression containing global variables or parameters in the Time step field.
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Specified values to store solutions at the values entered in the Times field in the General section.
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Steps taken by solver to store solutions at the time steps taken by the solver.
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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 Store reaction forces 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 Use lumping when computing fluxes 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 Store time derivatives 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 Store solution before and after events 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.
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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).
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The Black-Scholes Equation: Application Library path COMSOL_Multiphysics/Equation_Based/black_scholes_put
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