## Publications

Please see my Google scholar page for a list of my publications.

## Previous research

In the news at LBL: CRD Researchers Give Combustion System Design a Boost -- Optimized Algorithms Help Methane Flame Simulations Run 6x Faster on NERSC Supercomputer.### MLSDC + AMR

Spectral Deferred Correction (SDC) schemes are iterative methods for marching time-dependent problems through time. SDC methods construct high-order solutions within one timestep by iteratively approximating a series of correction equations at collocation nodes using low-order substepping methods. SDC schemes converge to the collocation solution (implicit Runge-Kutta schemes).

Multi-level SDC (MLSDC) schemes use a hierarchy of SDC schemes with varying number of collocation nodes to solve the collocation equation on the finest MLSDC level (ie, the level with the most SDC nodes) by cycling throught the MLSDC hierarchy in a V-cycle. Coarse and fine resolution SDC solutions on different MLSDC levels are coupled in the same manner as used in the full approximation scheme (FAS) method popular in multigrid methods for nonlinear problems.

For example, the collocation nodes of a three level MLSDC scheme with 3, 5, and 9 Gauss-Lobatto collocation nodes on the coarse, middle, and fine levels has the following node hierarchy:

I am currently developing an MLSDC+AMR solver for the multicomponent, compressible reacting Navier-Stokes equation called RNS. RNS will be fourth order accurate in both space and time and will showcase the MLSDC+AMR technique for combustion problems. A preliminary version of the code was used to create the movie below, which shows a volume rendering of the vorticity resulting from a turbulent jet. This simulation was run across several thousand cores of the Edison supercomputer at NERSC.

### Time-parallel schemes

My previous postdoc was at the University of North Carolina at
Chapel Hill under the supervision of
Michael Minion.
My research there was primarily focused on the *parallel full
approximation scheme in space and time* (PFASST) scheme for parallel-in-time integration
of PDEs.

PFASST is a novel approach to time parallelism that iteratively improves the solution on each time slice by applying deferred correction sweeps to a hierarchy of discretizations at different spatial and temporal resolutions. The coarse resolution problems are formulated using a time-space analog of the full approximation scheme used in multi-grid methods.

### High-order spatial reconstructions

I have collaborated with
David Ketcheson *et al.*
to incorporate high-order Weighted Essentially Non-oscillatory
(WENO) schemes into
PyClaw.

PyWENO (of which I am the primary author) was used to generate Fortran 90 routines that perform WENO reconstructions within PyClaw. The routines can perform WENO reconstructions from 5th to 17th order.

The PyWENO project provides a set of open source tools for constructing high-order Weighted Essentially Non-oscillatory (WENO) methods and performing high-order WENO reconstructions.

## Other research interests

My other research interests include:

- Numerical Analysis - Efficient implementation of Finite Volume schemes. Weighted Essentially Non-Oscillatory schemes for hyperbolic systems.
- Partial Differential Equations - Systems of hyperbolic conservation and balance laws, perturbation theory, Sobolev spaces, and weak solutions.
- Fluid Mechanics - Fluid dynamics, geophysical and environmental flows, gravity currents and sediment transport, free boundary flows and surface tension, turbulence, and applications in biology.
- Non-linear Dynamics and Chaos - Fixed point stability, bifurcations, and simple examples of the onset of chaos.
- Differentiable Manifolds - Hamiltonian mechanics, Lie groups, holonomic and non-holonomic reduction of constraints.
- Traffic Modeling - Incorporating stochastic phenomena into hyperbolic models of traffic flow.
- Dendrochronology - Analysing tree-ring width data to determine the time of death of dead trees.
- Population Dynamics - Modeling population dynamics using Integral Projection Models (IPMs).