From: Wolfgang Bangerth Date: Tue, 9 Jul 2024 22:19:10 +0000 (-0600) Subject: Leave another comment in the documentation of SUNDIALS::IDA. X-Git-Tag: v9.6.0-rc1~109^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=75bf8c3524cf1c8b62bf238ba54f69106ee06d9d;p=dealii.git Leave another comment in the documentation of SUNDIALS::IDA. --- diff --git a/include/deal.II/sundials/ida.h b/include/deal.II/sundials/ida.h index 555c5f2e78..a313acdcad 100644 --- a/include/deal.II/sundials/ida.h +++ b/include/deal.II/sundials/ida.h @@ -274,6 +274,31 @@ namespace SUNDIALS * variables is always in instantaneous equilibrium with another set of * variables that evolves on a slower time scale. * + * Another case where we *could* eliminate a variable but do not want to + * is where that additional variable is introduced in the first place to work + * around some other problem. As an example, consider the time dependent + * version of the biharmonic problem we consider in step-47 (as well as some + * later ones). The equations we would then be interested in would read + * @f{align*}{ + * \frac{\partial u(\mathbf x,t)}{\partial t} + \Delta^2 u(\mathbf x,t) &= + * f(\mathbf x,t). + * @f} + * As discussed in step-47, the difficulty is the presence of the fourth + * derivatives. One way in which one can address this is by introducing + * an auxiliary variable $v=\Delta u$ which would render the problem into + * the following one that only ever has second derivatives which we know + * how to deal with: + * @f{align*}{ + * \frac{\partial u(\mathbf x,t)}{\partial t} + \Delta v(\mathbf x,t) &= + * f(\mathbf x,t), + * \\ + * v(\mathbf x,t)-\Delta u(\mathbf x,t) &= 0. + * @f} + * Here, the introduction of the additional variable was voluntary, and + * could be undone, but we don't want that of course. Rather, we end + * up with a differential-algebraic equation because the equations do + * not have a time derivative for $v$. + * * Rather than show how to solve the trivial (linear) case above, let us * instead consider the situation where we introduce another variable $v$ that * is related to $u$ by the nonlinear relationship $v=u^p$, $p\ge 1$: