5.1.4. Implementation: Forward integration

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

See CellML syntax

<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