# Possible project topics¶

See Course Projects for general information about the course project.

There are many possible topics. I suggest using the Piazza discussion board to suggest topics if you are looking for someone to work with. Label your post with the project tag.

Ideally a project will require reading at least one paper from the literature and presenting a synopsis of the problem and solution, combined with some computations to support this. You are not expected to do original research as part of the project, but it should involve ideas, algorithms, and/or applications that go beyond what was covered in lectures and homeworks in some substantial way. It could focus on exploring a different class of algorithms and comparing to the wave-propagation algorithms we focused for some set of test problems, or applying the Clawpack software to some novel application (i.e. different than what’s in the Clawpack examples and apps, and that might require writing a new Riemann solver, for example).

There are several chapters of the book we aren’t covering in class, some of these could form the basis for a project.

There are many interesting things to study in one space dimension and several possible projects that include some discussion of theory, numerical methods, and applications that go beyond what is being covered in class, so it is fine to stick to a one-dimensional problem. But later in the quarter we will also get to multi-dimensional problems and methods and it’s also fine to tackle something in this direction.

Below are a few ideas, with some references. In many cases I’ve referred to my own papers since these tend to use similar notation and algorithms to what we will study in class, so these may be most accessible, particularly for students who are new to this field. But it would be great to explore the literature more broadly and work through papers by other authors. I’ve also given a couple examples of papers where interesting problems are solved with other methods and I think it would be interesting to compare with Clawpack. There are countless other such papers of this nature, so feel free to come up with papers on a different topic that you are interested in — you do not need to pick something from this list.

• Variable coefficient linear equations, e.g. acoustics or elasticity (perhaps in two space dimensions). Chapter 9 could be used as a starting point.

• Nonlinear equations with spatially varying fluxes $$f(q,x)$$. We will briefly discuss the “f-wave method” in class, but not in much detail and there are many things that could be tried. Layered nonlinear media is one interesting example where the reflections at interfaces give rise to dispersion, which couples with the nonlinearity to give solitary waves. Some references:

• Compare different approximate Riemann solvers for gas dynamics, e.g. “exact”, Roe, HLLE, perhaps consider other equations of state.

• Detonation waves in one dimension as an example of a conservation law with a stiff source term; see Sections 17.10 - 17.16, and the one-dimensional problem considered in this paper:

• Euler equations with “two-dimensional Riemann problem” data consisting of four constant states in the four quadrants of the plane. One such example is in the Clawpack gallery, but it would be nice to experiment with the numerous other examples presented in this paper:

• Explore the accuracy and efficiency of dimensional splitting vs. unsplit wave-propatation algorithms, both of which will be discussed to some extent in class. This is particularly interesting in three space dimensions where the unsplit algorithms are very expensive compared to dimensional splitting.

• The one-dimensional equations of magnetohydrodynamics (MHD), perhaps updating James Rossmanith’s MHDCLAW to the latest version of Clawpack and exploring this further.

• The wave-propagation methods we are studying are at most second order accurate. If higher-order polynomial reconstructions are used from the cell averages at each step, together with suitable limiters, it is possible to derive higher order shock-capturing methods, such as the class of WENO methods (Weighted Essentially Non-Oscillatory). There’s a big literature on these methods. One implementation is incorporated in PyClaw, but you might want to do your own to really understand these methods.

• The Buckley-Leverett equation discussed briefly in Chapter 16 is a nonconvex scalar equation. Explore multi-dimensional versions of this used in modeling multiphase flow in porous media (e.g. a subsurface oil reservoir).

• Chapter 16 contains some other nonclassical systems and might give you some ideas.

• A one-dimensional model of shock wave chaos, exploring the equation proposed in

• Exploring some more complicated models of traffic flow, e.g. variable speed limits (see Section 16.4.1), or modeling on-ramps (Section 17.11). There are many papers on other interesting traffic flow models, for example:

• Well-balanced methods for quasi-steady problems with source terms. This will be only briefly discussed in class.

• The chemotaxis problem in which bacteria move up the gradient of a nutrient. This is an advection-reaction-diffusion equation and might involve using a fractional step approach, e.g.

• The kitchen sink – Figure 13.3 shows a “cartoon” of a stationary radial shock wave in the shallow water equations. Properly solve this problem with radially symmetric 1d and/or full 2d equations.

• Moving mesh methods in one space dimension: In one dimension it is possible to define a fairly simple moving mesh method that can be used to study a contracting domain modeling a piston moving into a shock tube. This is described in the paper

The code that was used still exists but was written for an old verion of Clawpack (4.3) and might be interesting and useful to update this to Clawpack 5 form.