I am looking for a good resource for implementing adaptive mesh refinement (AMR) on a 1-D PDE and an analytical treatment. Pretty much everything I've found on AMR use 1-D as a toy example to show the benefits and then goes straight to 2-D problems, which are much harder mathematically and computationally for several reasons.
For a little background on why I would possibly need AMR for a 1-D problem, I want to solve the Cable equation, which models voltages in neurons. The Cable equation is the heat equation with a forcing term, and it is solved on a network of connected 1-D rods. (If you think of the typical artists rendition of a neuron, it is a branching network of cylindrical tubes.) Simulations are time consuming because there are usually a lot compartments in a neuron and a lot of neurons in a network, so a 50% speed up is a big deal.
I think AMR could really speed up the typical algorithm because in this paper, Rempe MJ, Spruston N, Kath WL, Chopp DL. Compartmental Neural Simulations with Spatial Adaptivity, they got around a 50% decrease in computation time simply by not simulating the parts of the neuron at steady state, no adaptive mesh. So if I used their idea along with an adaptive mesh, the decrease in computation time will be even better than 50%, which would be awesome!
So if anyone knows of a rigorous treatment of AMR in 1-D, along with good data structures to handle the changing 1-D mesh, please let me know. (I don't want to reinvent the wheel if I don't have to.) Also, I have a BS in computer science and an MS in applied math, so nothing in 1-D will go over my head.
Edit:
I was able to find this paper, [An adaptive mesh-moving and refinement procedure for one-dimensional conservation laws][1], but it was published in 1993. The paper has the analytical work I wanted, and a dynamic tree was used to handle the adaptive mesh.
I checked the citing papers as well, and there doesn't seem to be much work on 1-D adaptive mesh refinement. Maybe 1-D is too easy to justify much research, but I was hoping there was a more in depth look into which data structures work well with certain problems.
I will start digging in and answer this question once I have a satisfactory program.