Highlights include Computation’s annual external review, machine learning for ALE simulations, CFD modeling for low-carbon solutions, seismic modeling, and an in-line floating point compression tool.

# Topic: *Computational Math*

The code GEFIE-QUAD (gratings electric field integral equation on quadrilateral grids) is a first-principles simulation method to model the interaction of laser light with diffraction gratings, and to determine how grating imperfections can affect the performance of the compressor in a CPA laser system. GEFIE-QUAD gives scientists a powerful simulation tool to predict the performance of a realistic laser compressor.

Highlights include the HYPRE library, recent data science efforts, the IDEALS project, and the latest on the Exascale Computing Project.

The Extreme Resilient Discretization project (ExReDi) was established to address these challenges for algorithms common for fluid and plasma simulations.

Newly developed mathematical techniques reveal important tools for data mining analysis.

Livermore researchers have developed an algorithm for the numerical solution of a phase-field model of microstructure evolution in polycrystalline materials. The system of equations includes a local order parameter, a quaternion representation of local orientation, and species composition. The approach is based on a finite volume discretization and an implicit time-stepping algorithm. Recent developments have been focused on modeling solidification in binary alloys, coupled with CALPHAD methodology.

LLNL researchers are developing a truly scalable first-principles molecular dynamics algorithm with O(N) complexity and controllable accuracy, capable of simulating systems of sizes that were previously impossible with this degree of accuracy.

GLVis is a lightweight OpenGL-based tool for accurate and flexible finite element visualization. It is based on MFEM, a finite element library developed at LLNL. GLVis provides interactive visualizations of general finite element meshes and solutions, both in serial and in parallel. It encodes a large amount of parallel finite element domain-specific knowledge; e.g., it allows the user to view parallel meshes as one piece, but it also gives them the ability to isolate each component and observe it individually. It provides support for arbitrary high-order and NURBS meshes (NURBS allow more accurate geometric representation) and accepts multiple socket connections so that the user may have multiple fully-functional visualizations open at one time. GLVis can also run a batch sequence, or a series of commands, which gives the user precise control over visualizations and enables them to easily generate animations.

High-resolution finite volume methods are being developed for solving problems in complex phase space geometries, motivated by kinetic models of fusion plasmas.

LLNL researchers are testing and enhancing a neutral particle transport code and the algorithm on which the code relies to ensure that they successfully scale to larger and more complex computing systems.

LLNL and University of Utah researchers have developed an advanced, intuitive method for analyzing and visualizing complex data sets.

CASC researcher Carol Woodward consults on a diverse array of projects at the Laboratory and beyond. “It’s nice because it means I can work at the same place and not be stuck just doing one thing,” she says.

ODEPACK is a collection of Fortran solvers for the initial value problem for ordinary differential equation systems. The collection is suitable for both stiff and nonstiff systems and includes solvers for systems given in both explicit and linearly implicit forms.

The NSDE project is focused on research and development of nonlinear solvers and sensitivity analysis techniques for nonlinear, time-dependent, and steady-state partial differential equations.

The flourishing of simulation-based scientific discovery has also resulted in the emergence of the verification and validation (V&V) and uncertainty quantification (UQ) disciplines. The goal of these emerging disciplines is to enable scientists to make precise statements about the degree of confidence they have in their simulation-based predictions. Here we focus on the UQ discipline, which is essential for validating and verifying computer models.

A new algorithm for use with first-principles molecular dynamics codes enables the number of atoms simulated to be proportional to the number of processors available.

The Serpentine project develops advanced finite difference methods for solving hyperbolic wave propagation problems. Our approach is based on solving the governing equations in second order differential formulation using difference operators that satisfy the summation by parts principle.

Through research funded at LLNL, scientists have developed BLAST, a high-order finite element hydrodynamics research code that improves the accuracy of simulations, provides a path to extreme parallel computing and exascale architectures, and gives an HPC advantage.

The scalable multigrid reduction in time (XBraid) approach constructs coarse time grids and uses each coarse time scale solution to improve the next finer-scale solution, ultimately yielding an iterative scheme that simultaneously updates in parallel a solution guess over the entire space-time domain.