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Stiff ODEs with Jacobian Specification

Occasionally, you may come across a stiff differential equation, that is a differential equation that is numerically unstable and small variations in parameters cause different solutions to the ODEs. One way to tackle this is to choose a stiff-solver, or hybrid stiff solver (like the default LSODA). Typically this is enough. However exact Jacobian solutions may increase the stability of the ODE. (Note the Jacobian is the derivative of the ODE specification with respect to each variable). In rxode2 you can specify the Jacobian with the df(state)/dy(variable)= statement. A classic ODE that has stiff properties under various conditions is the Van der Pol differential equations.

In rxode2 these can be specified by the following:

## rxode2 2.1.2.9000 using 2 threads (see ?getRxThreads)
##   no cache: create with `rxCreateCache()`
Vtpol2 <- function() {
  ini({
    mu <- 1 ## nonstiff; 10 moderately stiff; 1000 stiff
  })
  model({
    d/dt(y)       <- dy
    d/dt(dy)      <- mu*(1-y^2)*dy - y
    ## Jacobian
    df(y)/dy(dy)  <- 1
    df(dy)/dy(y)  <- -2*dy*mu*y - 1
    df(dy)/dy(dy) <- mu*(1-y^2)
    ## Initial conditions
    y(0)          <- 2
    dy(0)         <- 0
  })
}

et <- et(0, 10, length.out=200) %>%
  et(amt=0)

s1 <- Vtpol2 %>%  solve(et, method="lsoda")
## i parameter labels from comments will be replaced by 'label()'
## using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’
print(s1)
## -- Solved rxode2 object --
## -- Parameters ($params): --
## mu 
##  1 
## -- Initial Conditions ($inits): --
##  y dy 
##  2  0 
## -- First part of data (object): --
## # A tibble: 200 x 3
##     time     y      dy
##    <dbl> <dbl>   <dbl>
## 1 0       2     0     
## 2 0.0503  2.00 -0.0933
## 3 0.101   1.99 -0.173 
## 4 0.151   1.98 -0.242 
## 5 0.201   1.97 -0.302 
## 6 0.251   1.95 -0.353 
## # i 194 more rows

While this is not stiff at mu=1, mu=1000 is a stiff system

s2 <- Vtpol2 %>%  solve(c(mu=1000), et)
## i parameter labels from comments will be replaced by 'label()'
## using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’
print(s2)
## -- Solved rxode2 object --
## -- Parameters ($params): --
##   mu 
## 1000 
## -- Initial Conditions ($inits): --
##  y dy 
##  2  0 
## -- First part of data (object): --
## # A tibble: 200 x 3
##     time     y        dy
##    <dbl> <dbl>     <dbl>
## 1 0       2     0       
## 2 0.0503  2.00 -0.000667
## 3 0.101   2.00 -0.000667
## 4 0.151   2.00 -0.000667
## 5 0.201   2.00 -0.000667
## 6 0.251   2.00 -0.000667
## # i 194 more rows

While this is easy enough to do, it is a bit tedious. If you have rxode2 setup appropriately, you can use the computer algebra system sympy to calculate the Jacobian automatically.

This is done by the rxode2 option calcJac option:

Vtpol <- function() {
  ini({
    mu <- 1 ## nonstiff; 10 moderately stiff; 1000 stiff
  })
  model({
    d/dt(y)       <- dy
    d/dt(dy)      <- mu*(1-y^2)*dy - y
    y(0)          <- 2
    dy(0)         <- 0
  })
}

Vtpol <- Vtpol()

# you can also use $symengineModelPrune if there is if/else blocks
# that need to be converted:
Vtpol <- rxode2(Vtpol$symengineModelNoPrune,  calcJac=TRUE)
## using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’
## > pruning branches (`if`/`else`)...
## v done
## > loading into symengine environment...
## v done
## > calculate jacobian
## [====|====|====|====|====|====|====|====|====|====] 0:00:00
## using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’
summary(Vtpol)
## rxode2 2.1.2.9000 model named rx_40d9bdb3df3f206a02e9833fed3f3d23 model (ready). 
## DLL: /tmp/RtmpdxI5WQ/rxode2/rx_40d9bdb3df3f206a02e9833fed3f3d23__.rxd/rx_40d9bdb3df3f206a02e9833fed3f3d23_.so
## NULL
## -- rxode2 Model Syntax --
## rxode2({
##     cmt(y)
##     cmt(dy)
##     d/dt(y) = dy
##     d/dt(dy) = -y + mu * dy * (1 - Rx_pow_di(y, 2))
##     y(0) = 2
##     dy(0) = 0
##     df(y)/dy(y) = 0
##     df(dy)/dy(y) = -1 - 2 * y * mu * dy
##     df(y)/dy(dy) = 1
##     df(dy)/dy(dy) = mu * (1 - Rx_pow_di(y, 2))
##     df(y)/dy(mu) = 0
##     df(dy)/dy(mu) = dy * (1 - Rx_pow_di(y, 2))
## })