Week 1
And we're off! Let the summer of math begin. These first days of the internship have been dedicated to getting ourselves oriented, both mathematically and otherwise. We moved into the room where we'll be spending most of the summer: Chapman 303C, on the campus of UAF. The three of us undergraduates have the room to ourselves, with desks, computers, and a blackboard with plenty of chalk.
After getting settled into our room, we started in on the math. Much of the first week was dedicated to studying sections of A Modern Introduction to the Mathematical Theory of Water Waves, a text that lays out the basics of using partial differential equations to model water waves, and a set of notes on the subject given to us by Professor Rybkin. I arrived on Friday after returning late Thursday night from an out-of-town ecology course, so while Jeremiah and Matthew had already moved on to researching related topics, I dug into the the material they'd already covered.
Week 2
Throughout the second week, we have still been finding our feet. Professor Rybkin was away on a trip all week, so we have been more or less self-directed, with some very helpful hints from Odile and Dmitry. As the week started off, we dove back into the notes that Professor Rybkin had given us, each of us going through them separately and discussing points that we didn't understand or disagreed upon. After a couple days of this, we felt like we finally had a handle on what was going on, and were thus ready to move on.
At the conclusion of Professor Rybkin's notes, a partial differential equation was derived to model the wave in question, and in fact is in a form well-known enough to merit it's own name: the Klein-Gordon equation, a special case of the Schrödinger equation. However, this equation is no longer using the same units or axes as the initial problem was framed in, so in order to solve it, we will need to pull the initial physical data through a long series of transformations to end up in the variables the Klein-Gordon equation uses.
And then, in order to solve the Klein-Gordon equation, we will need to use techniques from functional analysis, including spectral theory. None of us undergraduates have had experience with functional or spectral analysis, so I headed over to the library to check out a list of books we had decided would be useful.
Meanwhile, Dmitry pointed out that once we solved the Klein-Gordon equation, the solution would still be in extremely abstract variables, and that we would need to have some way to back-substitute all the way to physical variables. He suggested that we look at a technique first used in a 1958 paper on the mathematical modeling of water waves written by G.F. Carrier and H.P. Greenspan. So the last part of this week has been dedicated to reading that and related papers, and working on applying what we've learned to our specific equations.
Finally, this summer's website is finally live on the internet! However, it is still existing in its own little world, as no other pages yet link to it. Once I have finished adding content to some of the other pages, I'll add a link from the main REU homepage and break this little site's isolation.
Week 3
We spent most of this week wrapping up the review work we had to do in order to have the tools to tackle this summer's problem. I spent the early part of the week working on a certain aspect of the equations we would use to back-substitute to the physical variables. Matt worked on using MATLAB computer software to get numerical approximations of the Schrödinger Potential given an arbitrary bay cross-section, as we were able to analytically solve for it in only a few cases. Jeremiah took some work I had done last week on the Schrödinger Potential in certain cases of U-shaped bays and generalized it for any arbitrary bay whose shape could be approximated using a power function.
On Wednesday, following on a suggestion from Dr. Rybkin, Jeremiah and I turned our attention to Fourier and Fourier-like transformations, collectively known as integral transformations. When used in the context of differential equations, these allow a partial differential equation in two variables to be transformed into an ordinary differential equation in one variable. In all the previous work done on water wave modeling, Fourier-like transformations were used in the process of finding the solution, and we expect that they will be involved in ours as well. Thus, we wanted to be fully familiar with these transformations.
Matt, who had already had quite a bit of experience with Fourier and similar transforms, spent the tail end of the week studying a recent draft of an unpublished paper by Dr. Rybkin, Dr. Pelinovsky, and Dr. Didenkulova, all leading scholars in the world of water wave simulation. Parts of this paper include the setup for the problem we will be working on for the rest of the summer: modeling tsunami run-up in a bay with a trapezoidal cross-section. On Friday, once Jeremiah and I had gotten more familiar with integral transformations, we read through this paper as well, in order to familiarize ourselves with the specific notation with which we will be working.
Week 4
This week, we began examining the specific details of the case we will be attempting to model: that of a wave entering a bay with trapezoidal cross-section. We immediately ran into problems: after quite a bit of work, we found that the solution to a key integral in the process of converting our physical variables x and t to non-physical variables σ and λ can't be written in elementary functions. This is a major stumbling-block, and we'll need to find some way around it if we want to proceed. Jeremiah spent a good portion of this week looking to see if there was some way to transform the integral (through Laplace Transforms and such) into a form that would give a better solution, but he didn't have any luck. Matt took a more direct route, and focused on finding a way to numerically integrate without getting a lot of error around σ = 0. After several days of work, a regularization of the integral was found, and Matt was able to numerically integrate that to get reasonably low error.
Meanwhile, I was investigating properties of several quantities that we believed we'd have to work with. In the notes that we started the program with, there was some ending discussion on an operator, L, that could potentially be analyzed using spectral analysis, so I spent some time and found its eigenfunctions. Then Matt told me that even once we found W(σ), a critical term in the differential equation for Φ, we would need its derivative. So I used some chain rule techniques to work out a direct expression for its derivative. Finally, I found an algebraic expression for q(H), the Schrödinger Potential in terms of the σ(H) that Matt and Jeremiah were trying to find. Once Matt had found his numeric approximation, we plugged it into the expression I had derived and we had an approximate q. We finished the week off with a group discussion with Dr. Rybkin and Dmitry.
Week 5
This was a shorter work week, as the 4th of July fell on Wednesday, and we were given Thursday off was well. Monday and Tuesday were mostly dedicated to reading and studying a very recent (published in 2011) paper by Dr. Didenkulova and Dr. Pelinovsky entitled "Nonlinear wave evolution and runup in an inclined channel of a parabolic cross-section." We hope that techniques used in this paper in the parabolic case will help us crack the trapezoidal one. Dr. Rybkin also had a new set of notes for us that we verified and studied. Not much more was done this week, other than, everyone having a great 4th of July!
Week 6
This week, the focus was on numerical techniques. Matt started working on a numerical solver for our PDE using central difference schemes, and Jeremiah was working on a numerical process of transforming our abstract variables back to physical ones, once we had solved the equation. Eventually, Matt took over the numerical backsubstitution code, while Jeremiah started working on proving that such a backsubstitution was well-defined.
I started out the week by looking back at a remark in the book Partial Differential Equations for Scientists and Engineers, which was one of the books Dr. Rybkin recommended we read prior to starting the program. He had noticed an error in the remark, and he was curious if with our new experience working with PDEs whether we could identify it. The remark stated that a technique for solving a constant-coefficient system could be generalized to a non-constant-coefficient system, which would have been of great utility to us if it were true. Unfortunately, it wasn't, because the author neglected to apply the product rule.
I then investigated a few potential techniques for solving our technique, mainly focusing on using the Laplace transform of our differential equation. After that didn't yield any direct results, I looked at using Picard iteration on the transformed equation. Once again, this didn't lead to any meaningful new information.
On Friday, we headed out for a weekend trip down to see the Root Glacier at McCarthy, Alaska. McCarthy is extremely isolated, tucked into the middle of the Wrangell-St. Elias National Wilderness Preserve at the end of 60 miles of unpaved road. Friday was mostly consumed by driving (over 9 hours from Fairbanks!) but on Saturday, we did two hikes. On the first hike, we walked out onto the glacier itself, getting fantastic views up at the mountains all around, and on the second hike we climbed one of said mountains. We hiked up to the abandoned Bonanza mine, a 9 mile hike with 4000 feet of elevation gain. Exhausting as it was, the hike was well worth it, as we got fantastic views of nearly the entire length of the glacier and the surrounding mountain range. Click here to see some photos from the trip.
Week 7
This week, we made some real progress. The numerical approximation program using implicit finite differences that Matt and Jeremiah had been working on began to produce results that looked encouraging, providing we were able to approximate a crucial parameter W. I spent most of the week working with Matt and Jeremiah on the approximation program, and getting up to speed with the MATLAB code.
The primary issue in approximating W is that as σ goes to 0, which is at the moving shoreline where it is most important to get accurate results, the error in any numerical approximation of W becomes large, since W contains a 1/σ term. So rather than using a numerical approximation near 0, the grad student with whom we are working derived the asymptotics for W, so I spent some time later in the week verifying his results.
I also spent some time attempting to see if our fully transformed, linearized equation in abstract variables was equivalent to the classic wave problem, although I was unable to fully verify this.
Week 8
Our model finally came together! Matthew spent some time optimizing the step size in the finite difference model to minimize error, at least in the parabolic case where we have the exact data to allow us to calculate error. He found that a λ-step of .001 and a σ-step of .01 yields a model that runs in about a minute and has max error of approximately 2% at the shore.
Meanwhile, Jeremiah finished up the backsubstitution code so that when we found a solution to our equation in σ and λ back to the physical variables x and t. He also corrected a mistake I had made earlier in the program, when I thought that it was impossible to analytically find the integral of F(σ); Jeremiah showed instead that this integral is always equal to 2gH.
My main contribution this week was working with Jeremiah to create a program that numerically estimated F, and thus W, for an arbitrary bay shape. I then took that program and modified it so that it would smooth out some of the roughness from the numerical error, and then stitch it together with the asymptotic results that we had for F and W to get a reasonable approximation across σ.
Week 9
This last week of the program was short, but very productive. On Monday, Jeremiah and I finished the F-finder program, and included an algorithm that let the program dynamically stitch together the asymptotics and the numerical estimation for F. Matt continued optimizing the numerical model, and towards the end of the day we started integrating the two programs.
Tuesday, we finally got results. The programs were fully integrated, so we could feed in a bay shape and initial profile, the F-finder program would estimate F and W, the numerical PDE solver would give us results in the non-physical variables, and the back-substitution program took that data and put it into useable form. An example of what the program outputted is here, where what is being displayed is a vertical cross-section of the wave rolling in along the centerline of the bay. So the blue line is the shoreline profile. Something interesting to note is that at certain points, the line for the solution kinks slightly, which is the result of the stitching between the asymptotic F and numerical F not being completely smooth. It is a problem, but we hope that it won't affect the data too much.
Wednesday was the last day of the program. We handed off all of our code and results to Dr. Rybkin and Dr. Nicolsky, packed up our desks, and started planning for the final presentation that we'll have to do later this year. (Edit: Here is a copy of that presentation.) And then Dr. Rybkin, his wife, Matt, his girlfriend, and I all headed out of town to Cordova on a trip that Dr. Rybkin had planned. We spent a couple days in Cordova, the highlight of which was a trip up to Child's glacier, a few mile boat trip up the Copper River. We spent a few hours on the bank of the river, and got to see several excellent calving events and the waves they generated, which was the relevance to our program. Some pictures from the trip can be found here.
And this is the end of my blog. This summer was a fantastic experience during which I learned a ton about mathematics, and research in general. There are a few weeks left before the semester begins, and Dmitry has asked me to accompany him on a trip to Cold Bay and King Cove, Alaska on a GPS mapping trip, so the fun is just beginning!