Description: This example shows two possible strategies to handle resets. No particular method is prescribed (just as no method is prescribed for the solution of ODEs either).
Note that:
- all elements are in the same component;
- the order values of resets are not shown; and
- the units of variables are not shown.
component: ForwardIntegration
├─ math:
│ └─ ode(A, t) = 1
│
└─ variable: A initially 1
└─ reset:
├─ when A == 3
└─ then A = 1
<variable name="t" units="dimensionless" />
<variable name="A" units="dimensionless" initial_value="1" />
<!-- Reset rule 1: -->
<reset variable="A" test_variable="A">
<test_value>
<cn units="cellml:dimensionless">3</cn>
</test_value>
<reset_value>
<cn units="cellml:dimensionless">1</cn>
</reset_value>
</reset>Near t = 1.9, an integration step proceeds as normal: the derivatives f(x_1, t_1) are calculated, and used to estimate a step (x_2, t_2).
The quantity A - 3 (test variable minus test value) is calculated and monitored for sign changes.
| t | 1.9 | Propose: 2.1 | |
| A | 2.9 | Propose: 3.1 | |
| A - 3 | -0.1 | Propose: 0.1 |
After this step is proposed, the test variables are checked. Because the sign of A - 3 has changed, the integrator notices a discontinuity has occurred, and so the reset should be carried out as soon as possible.
To do this, the reset value is evaluated at (x_1, t_1) - just like the derivatives were evaluated at (x_1, t_1) earlier - and x_2 is updated to reflect this change:
| t | 1.9 | Propose: 2.1 | Accept: 2.1 |
| A | 2.9 | Propose: 3.1 | Accept: 1 |
| A - 3 | -0.1 | Propose: 0.1 |
Variable time-step methods can try and find the discontinuity, for example they could use threshold crossing algorithms to find a better proposed step:
| t | 1.9 | Propose: 2.1 | Update: 2.0 | Accept: 2.0 |
| A | 2.9 | Propose: 3.1 | Update: 3 | Accept: 1 |
| A - 3 | -0.1 | Propose: 0.1 | Update: 0 |