Differential Algebraic Equations

Differential Algebraic Equations (DAEs) are differential equations which have constraint equations on their evolution. This tutorial will introduce you to the functionality for solving differential algebraic equations (DAEs). Other introductions can be found by checking out SciMLTutorials.jl.

Note

This tutorial assumes you have read the Ordinary Differential Equations tutorial.

Mass-Matrix Differential-Algebraic Equations (DAEs)

Instead of just defining an ODE as $u' = f(u,p,t)$, it can be common to express the differential equation in the form with a mass matrix:

$$$Mu' = f(u,p,t)$$$

where $M$ is known as the mass matrix. Let's solve the Robertson equation. In previous tutorials we wrote this equation as:

\begin{aligned} dy_1 &= -0.04 y_1 + 10^4 y_2 y_3 \\ dy_2 &= 0.04 y_1 - 10^4 y_2 y_3 - 3*10^7 y_{2}^2 \\ dy_3 &= 3*10^7 y_{2}^2 \\ \end{aligned}

But we can instead write this with a conservation relation:

\begin{aligned} \frac{dy_1}{dt} &= -0.04 y_1 + 10^4 y_2 y_3 \\ \frac{dy_2}{dt} &= 0.04 y_1 - 10^4 y_2 y_3 - 3*10^7 y_{2}^2 \\ 1 &= y_{1} + y_{2} + y_{3} \\ \end{aligned}

In this form, we can write this as a mass matrix ODE where $M$ is singular (this is another form of a differential-algebraic equation (DAE)). Here, the last row of M is just zero. We can implement this form as:

using DifferentialEquations
function rober(du,u,p,t)
y₁,y₂,y₃ = u
k₁,k₂,k₃ = p
du[1] = -k₁*y₁ + k₃*y₂*y₃
du[2] =  k₁*y₁ - k₃*y₂*y₃ - k₂*y₂^2
du[3] =  y₁ + y₂ + y₃ - 1
nothing
end
M = [1. 0  0
0  1. 0
0  0  0]
f = ODEFunction(rober,mass_matrix=M)
prob_mm = ODEProblem(f,[1.0,0.0,0.0],(0.0,1e5),(0.04,3e7,1e4))
sol = solve(prob_mm,Rodas5(),reltol=1e-8,abstol=1e-8)

plot(sol, xscale=:log10, tspan=(1e-6, 1e5), layout=(3,1))

Note

If your mass matrix is singular, i.e. your system is a DAE, then you need to make sure you choose a solver that is compatible with DAEs

Implicitly-Defined Differential-Algebraic Equations (DAEs)

In this example we will solve the Robertson equation in its implicit form:

$$$f(du,u,p,t) = 0$$$

This equation is a DAE of the form:

\begin{aligned} \frac{du}{dt} &= f(u,p,t) \\ 0 &= g(u,p,t) \\ \end{aligned}

which is also known as a constrained differential equation, where g is the constraint equation. The Robertson model can be written in the form:

\begin{aligned} \frac{dy_1}{dt} &= -0.04y₁ + 10^4 y_2 y_3 \\ \frac{dy_2}{dt} &= 0.04 y_1 - 10^4 y_2 y_3 - 3*10^7 y_{2}^2 \\ 1 &= y_{1} + y_{2} + y_{3} \\ \end{aligned}

with initial conditions $y_1(0) = 1$, $y_2(0) = 0$, $y_3(0) = 0$, $dy_1 = - 0.04$, $dy_2 = 0.04$, and $dy_3 = 0.0$.

The workflow for DAEs is the same as for the other types of equations, where all you need to know is how to define the problem. A DAEProblem is specified by defining an in-place update f(out,du,u,p,t) which uses the values to mutate out as the output. To makes this into a DAE, we move all of the variables to one side. Thus, we can define the function:

function f(out,du,u,p,t)
out[1] = - 0.04u[1]              + 1e4*u[2]*u[3] - du[1]
out[2] = + 0.04u[1] - 3e7*u[2]^2 - 1e4*u[2]*u[3] - du[2]
out[3] = u[1] + u[2] + u[3] - 1.0
end

with initial conditions

u₀ = [1.0, 0, 0]
du₀ = [-0.04, 0.04, 0.0]
tspan = (0.0,100000.0)

and make the DAEProblem:

using DifferentialEquations
differential_vars = [true,true,false]
prob = DAEProblem(f,du₀,u₀,tspan,differential_vars=differential_vars)

differential_vars is an option which states which of the variables are differential, i.e. not purely algebraic (which means that their derivative shows up in the residual equations). This is required for the algorithm to be able to find consistent initial conditions. Notice that the first two variables are determined by their changes, but the last is simply determined by the conservation equation. Thus, we use differential_vars = [true,true,false].

As with the other DifferentialEquations problems, the commands are then to solve and plot. Here we will use the IDA solver from Sundials:

using Sundials
sol = solve(prob,IDA())

In order to clearly see all the features of this solution, it should be plotted on a logarithmic scale. We'll also plot each on a different subplot, to allow scaling the y-axis appropriately.

using Plots
plot(sol, xscale=:log10, tspan=(1e-6, 1e5), layout=(3,1))

This gives the following plot: