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DiscreteVariables

DiscreteVariables

is an option for NDSolve and other functions that specifies variables that only change at discrete times in a temporal integration.

Details

DiscreteVariables->{v₁, v₂,…} specifies that v₁, v₂, … should be treated as variables that only change at discrete times.
The value of a discrete variable v can be changed through WhenEvent[event,v->val] or WhenEvent[event,v->"DiscontinuitySignature"].
The discrete variable solution v can be returned by using NDSolve[eqn,{v,…},…].
DiscreteVariables->{vspec₁,vspec₂,…} can be used to specify the ranges for discrete variables.
Possible forms for vspec_i are:

	v	v has range Reals or Complexes
	Element[v,Reals]	v has range Reals
	Element[v,Complexes]	v has range Complexes
	Element[v,Integers]	v has range Integers
	Element[v,{n₁,…}]	v has discrete range {n₁,…}
	{v,v_min,v_max}	v has range
	vspec_iaction_i	perform action_i when vspec_i is no longer satisfied

Derivatives with respect to time of discrete variables are zero almost everywhere and should not appear in the equations.
For partial differential equations, discrete variables can depend only on the temporal independent variable in a method of lines semi-discretization.

Examples

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Basic Examples (2)

Solve for with , a discrete variable that changes at regular intervals in time:

Solve for with , a discrete variable that changes whenever crosses 0:

Scope (7)

Increment a discrete variable at regular time intervals:

Increment a discrete variable when the solution crosses 0:

Modify multiple discrete variables simultaneously when the solution crosses 0:

This time, enact the change in before modifying :

Stop the integration when a discrete variable goes out of the discrete range {1,2,3}:

Stop when the discrete variable goes out of the continuous range :

Print a message when out of range, but continue integrating the equation:

The initial condition is also out of range:

Discrete variables can take on non-numerical values:

Allow sliding mode solutions by using the action "DiscontinuitySignature":

Plot the vector field and solution:

The value of the discrete variable is 0 when the solution is in sliding mode:

Set the discontinuity state variable when reaches a sliding discontinuity curve :

Applications (5)

Switch between two right sides of a differential equation using a discrete variable:

Set up a differential equation that switches between multiple right sides:

Simulate a ball bouncing down steps:

Plot the ball's kinetic, potential, and total energy:

Simulate the system stabilized with a discrete-time controller :

Change the wave speed at in a wave equation:

Properties & Relations (1)

NDSolve automatically handles discontinuous functions like Sign using discrete variables:

Use "DiscontinuitySignature" with a discrete variable to emulate the Sign function:

Possible Issues (3)

When a discrete variable goes out of range, a message is displayed and the integration halts:

Derivatives of discrete variables cannot appear in the equations passed to NDSolve:

Discrete variables with "DiscontinuitySignature" action must have range {-1,0,1}:

If the range is {-1,1}, the sliding mode solution will not be found:

Specify the range as Element[a,{-1,0,1}] for sliding mode solutions:

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