# Computational and Applied Mathematics at Boise State University

The Computational and Applied Mathematics (CAM) Group consists of Stephen Brill, Donna Calhoun, Michal Kopera, Jodi Mead, Grady Wright, and Barbara Zubik-Kowal.

We are pleased to announce a new PhD in Computing with emphases in Computational Math Science & Engineering (CMSE),Â and in Data Science. These degree programs will be supported by our Computational and Applied Math faculty, and we have we many research projects that align well with the program.

## Faculty interests

Stephen Brill‘s research is in the area of numerical solution of ordinary and partial differential equations, particularly those which model single- and multi-phase flow and contaminant transport in porous media. He is presently working on obtaining analytical formulas for collocation discretizations of convection-diffusion equations and studying these formulas to choose the value of free parameters so as to obtain highly accurate numerical solutions.

Donna Calhoun works on solving partial differential equations using finite volume methods on logically Cartesian meshes. Two areas are of particular importance when using Cartesian meshes are handling problems that are posed in non-rectangular domains, and efficiently distributing computational resources only in regions of the Cartesian mesh where the solution of interest. To handle geometry, Donna has developed immersed interface methods, embedded boundary methods and mapped grid methods. Immersed or embedded boundary methods simply cut complex geometry out of the background Cartesian mesh, and treat the irregular cells near the embedded boundary as special cases. In mapped grid approaches, the Cartesian mesh is mapped, via a smooth or piecewise smooth transformation to a non-rectangular domain such as a disk or surface mesh. For all of these methods, special finite volume solvers must be developed for the hyperbolic, elliptic or parabolic terms the equations of interest. To improve computational efficiency, Donna has also worked extensively with adaptively refined meshes (AMR).

Michal Kopera is interested in computational and applied mathematics, high-performance scientific computing, computational fluid dynamics, adaptive mesh refinement, and scientific software development. Michal is working on developing ocean models using modern numerical methods (spectral elements, discontinuous Galerkin). An important aspect of his work is the ability of a model to represent complex geometries, and dynamically adapt the mesh to an evolving solution.

Grady Wright‘s research interests are in high-order methods for partial differential equations, approximation theory, low rank methods, scientific computing, and numerical software development.Â He works onÂ problems inÂ biology (biofluids and biomechanics), geophysics (geophysical fluid flows), and astronomy (cosmic microwave background).Â He develops methods based on radial basis functions, (compact) finite-differences, and spectral methods for these problems. A common theme ofÂ his work is complex geometries, such as spheres and more general surfaces.

Jodi Mead‘s work relates to the mathematical theory of problems from engineering and the natural sciences, including their computer solutions. This involves ordinary and partial differential equations, inverse theory, and parallel algorithms. Some specific applications she has worked on are in computational aeroacoustics, ocean modeling, near-surface geophysics and hydrology. Her work has included numerical solution of differential equations over large time and spatial scales, differential equations in Lagrangian form, and the blending of numerical solutions of mathematical models with different types of large data sets.

Barbara Zubik-Kowal is working on a variety of problems: cancer models, immune system dynamics, threshold models, dendritic and brain models, models in fluid mechanics, chaos, electromagnetics, space-time dependent NLS with memory, and others. The model problems are described by delay differential equations (DDEs) – a general class including both delay and classical ordinary and partial DEs, which depend additionally on their solutions at some past stage(s). DDEs are popular as they arise from various applications, like biology, medicine, physics, control theory, and others. Her research activities focus on predictive modeling, parallel scientific computing (delay, integro-differential and classical DEs), numerical stability, construction and implementation of novel numerical methods, and development of numerical software.