Time step index

Introduction

Specifying different time step indices for different submodels is an advanced topic. It can lead to misleading results, and should be used only in accordance with the following notes.

By default, all the model state variables are updated using the same time step. When Euler integration is used to update the state variables, differential equations are, in practice, solved as if they were difference equations, with a small but finite δt. By specifying δt , the time step, to be reasonable small, the results of the difference equations are the same as the expected results of the differential equations. You can, however, take advantage of this ability to specify δt, to use different time steps for different submodels.

A common example is in the calculation of compound interest on money in a savings account. It is important when specifying that interest is to be compounded, that the period over which the compounding occurs is specified. For example, the same investment in two savings accounts, both paying compound interest at 12 per cent, will yield different amounts after one year if one account compounds monthly and the other compounds annually. In modelling these accounts, you would need to specify a different δt, the time step, for each, in order that the state variable (the amount in the account) was updated at different rates, either monthly or annually.

Another reason for having several time steps is when a model contains components that behave on different time scales. For instance a model of a grazing system might contain a lot of plants, with physiological processes that can be modeled with a time step of a day, and animals whose movements would be modeled on a time step of seconds or minutes. To model all the plants at the shorter time step would be inefficient, so the animals are handled by a separate submodel which is run with a shorter time step.

It is possible to specify up to seven time step indices. For each new time step category that you request, an extra Time step #x entry is added in the Run Control window, and that is where you specify the actual time step (e.g. 0.01) to be used for each index.

Notes

The following notes will help you use this concept:

• Note that the TIME UNIT is the same over the whole model, and that all flows are expressed per time unit. The run control also specifies how long the model should be run for, in terms of time units.

• Each submodel TIME STEP is specified in terms of a multiple of the time unit. The submodel time steps *must* be ordered from long to short, #1 must be the longest, #7 the shortest. During each time step that affects a submodel (i.e. its own and all longer ones) all the elements of the submodel are updated. Note, particularly, that since flows are expressed in time units, a compartment's value does not change each time step by the value that is entered (or calculated) for the flow, but by the same multiple of that value as the time step is of the time unit.

• To refer to the time step in the model, use the function dt().

Special cases

In addition to the seven time step indices available, there are three special cases:

• "Initialize only" denotes that the submodel should be updated only when the model is built.
• "New params only" denotes that the submodel should be updated only when the model is built or when it is reset after new values for fixed parameters have been loaded.
• "Reset only" denotes that the submodel should be updated only when the model is built or when it is reset.

In none of these cases will the submodel be updated during the simulation.

Example

For the example of compound interest given above, set the time step index for the annual submodel to #1 and for the monthly submodel to #2. In the run control, you will then need to set time step #1 = 12 and time step #2 = 1. Flows in both submodels will be expressed in the same time units, in this case, per month.

Note that this is exactly equivalent to time step #1 = 1 and time step #2 = (1/12). Then the implicit time unit is one year, and the flows will be expressed on an annual basis.

As a final point, note that it is possible to set explicitly the time unit for each flow. Enter explicit physical units such as kg/week or kg/day. If the compartment has compatible physical units, such as kg, then the appropriate conversion will be performed. If you do this, note especially that you must be consistent throughout a network of flows.

In: Contents >> Working with submodels >> Submodel properties