I would guess they are solving PDEs using systems of linear equations. The approximate solutions to these linear equations are found via the analog linear solvers.
For others who want a comparison; the two papers are very different: the Nature paper uses entirely analog techniques to achieve high precision results while the newly-linked article is a mixed-precision approach. That is, the newly-linked article only uses analog to obtain a low-precision estimate which is fed back to a digital computing algorithm to produce the final high-precision result. The hope is that the overall time to solution is still faster because you do not need (hopefully eventually slower) digital computation to achieve the low-accuracy result. I am not an expert in this particular hardware field (I do applied math so I am used to solving the problems they do here in digital computers), but here is what the authors of the Nature article say:
> Recent theoretical and experimental studies have used memristor (including phase-change memory) arrays to generate an initial, low-precision guess (seed), and rely on an integrated high-precision digital solver to produce the required high-precision solutions from the seed solution. Such types of accelerators are certainly beneficial, as they reduce the number of iterations required by the digital solver. Determining whether memristor-based hardware can be used to directly perform high-precision computing tasks, however, will enable a better understanding of how broadly memristive hardware can be applied. Such knowledge will help pave the way to build more general memristor-based computing platforms, instead of special-purpose accelerators.
However, both of those problems do include a subproblem which is solving linear systems (when the ODE is solved implicitly), and so PDE solvers which take these routes will generally be able to be accelerated by an improved linear equation solver. Thus it's not an be all and end all algorithm, but when combined with the tools that exist for PDEs in something like PETSc or Julia then this can be used to give a more efficient algorithm for most PDEs (though there are still other PDEs where linear solving is entirely avoided, for example with exponential integrators or Runge-Kutta Chebyshev methods, which can be more efficient depending on the problem).
You're correct that the innovation is really about sparse linear algebra (though not just solvers) done in hardware. However the advantages are exploited not just by way of solving linear systems for implicit time-stepping. They implemented a Jacobi iteration in hardware for solving a system arising from discretization of the Poisson equation, but they also used an efficient forward-map of the coefficient matrix to do explicit time-stepping for a wave solver. So they solve both a time-independent and time-dependent problem, but neither with implicit time-stepping (though there appears to be no technical limitation here, it's just what they focused on).
It even has a .pdf extension.
This crashed my browser.