The Association for Women in Mathematics (AWM) was founded in 1971, to encourage women to have careers in mathematical sciences. They held a research collaboration workshop in August 2014, hosted by the Institute for Mathematics and its Applications (IMA). This book contains six papers, by six groups, on numerical partial differential equations. Each group consists of at least one senior mathematician.

The first paper discusses a numerical method for solving elliptic optimal control problems with point-wise state constraints in a bounded convex polyhedral domain. The authors show that continuous interior penalty methods [1] are effective for the solution of the problem with either Dirichlet or Neumann boundary conditions. The method has been previously applied only to 2D domains. The authors describe three different post-processing methods to obtain approximate optimal control from the approximate optimal state, and give six examples showing linear rate of convergence for the optimal state and second order in the energy norm. They also observe that the last two post-processing procedures yield better results than the first one.

The second paper deals with the sensitivity parameter in weighted essentially non-oscillatory (WENO) methods [2] to capture shocks in the solution of 1D hyperbolic problems in conservation law form. They show that the parameter used by WENO methods initially to avoid division by zero should not drop below a certain critical value. This critical value depends on the size of the jump discontinuity and the function used. The authors compare three WENO procedures of order five for smooth problems and third order in non-smooth regions. The system of first-order ordinary differential equations is solved using Runge-Kutta methods. Several linear and nonlinear examples are given for comparison of the three WENO schemes.

In the third paper, the authors study a population dynamics model. This is described by an integro-differential equation, which is first order in time and second order in space. The kernel is smooth, monotonically decreasing function with compact support. The boundary conditions considered are either Dirichlet or periodic. The authors are interested in finding regions where the species will die out more slowly or survive more easily. Several analytical results and finite difference approximation of the mathematical model in 1D and 2D are given. Several numerical experiments are detailed.

A decoupling algorithm for a system of fluid and poroelastic material intersection is the topic of the fourth paper. As an example, consider the blood flow interaction with the arteries [3]. Decoupling allows the solution in parallel of each subproblem. The problem is cast as a constrained optimization problem where the violation of the interface conditions is to be minimized. According to the paper, “two numerical algorithms based on a residual updating technique are” used and compared in an example.

The next paper studies “linear kinetic models arising from semiconductor device simulations,” for example, 1D electron-phonon scattering. The authors “investigate how various ... numerical methods approximate [the] scattering operators.” The methods are first- and second-order discontinuous Galerkin, first-order collocation, the Fourier-collocation spectral method, and the Nyström method. Numerical experiments comparing these methods are given. It would be very interesting to extend the work to higher dimensions and theoretically analyze the methods.

The last paper deals with coupling in multiphysics simulations. In the literature, physics is decoupled to make the problem tractable. This is done via operator splitting [4]. The authors suggest a metric to measure how strong the coupling is. In cases where the coupling is strong, a smaller time step will be used to better incorporate the physics. Four different measures are considered and tested on diffusion-reaction problems (linear as well as nonlinear). It is demonstrated that the metrics based on Jacobin are not as good of a predictor as the one based on time scales.