A Schrodinger equation for the system's wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials' properties. Recent studies have demonstrated how enriched finite element type Galerkin methods can substantially lower the number of degrees of freedom necessary to produce accurate solutions with respect to the standard plane-waves method. In particular, the flat-top partition of unity finite element method enriched with the radial eigenfunctions of the one-dimensional Schrodinger equation offers a very effective way of solving the three-dimensional Schrodinger eigenvalue problem. We investigate the theoretical properties of this approximation method, its well-posedness and stability, we prove its convergence and derive suitable bound for the h- and p-refinement in the L-2 and energy norm for both the eigenvalues and the eigenfunctions. Finally, we confirm these theoretical results by applying this method to the eigenvalue problem of the one-electron Schrodinger equation with the harmonic potential, for which the exact solution is known.

We introduce featom, an open source code that implements a high-order finite element solver for the radial Schrodinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the ������ = +/- 1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10-8 Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schrodinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (������) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines.Program summaryProgram Title: featomCPC Library link to program files: https://doi .org /10 .17632 /962fzmm7f7 .1Licensing provisions: MITProgramming language: Fortran with command-line interfaces Nature of problem: Solution of the Schrodinger, Dirac, and Kohn-Sham equations of density functional theory for isolated atoms. Solution method: A high-order finite element method is used to assemble a generalized eigenproblem that is then solved for eigenvalues and orbitals. The Poisson equation is solved using the finite element method also.Self-consistent field equations are solved using Pulay mixing. Additional comments including restrictions and unusual features: Our solution methodology for the Dirac equation does not suffer from spurious states. We maintain high accuracy even for ������ = +/- 1 states. Unlike other methods, our approach is not limited to Coulombic or self-consistent potentials and can handle non-uniform meshes, including exponential meshes. Restrictions: Spherical symmetry.

— A community of developers has formed to modernize the Fortran ecosystem. In this article, we describe the high-level features of Fortran that continue to make it a good choice for scientists and engineers in the 21st century. Ongoing efforts include the development of a Fortran standard library and package manager, the fostering of a friendly and welcoming online community, improved compiler support, and language feature development. The lessons learned are common across contemporary programming languages and help reduce the learning curve and increase adoption of Fortran.

We present a new Fortran source-to-source tool that helps to bridge the gap in the current Fortran tooling ecosystem. The goal of this tool is to translate standard Fortran code to various parallel programming languages in Fortran and C/C++ to enable running on a wide variety of GPUs and CPUs. The translation is performed using the full syntax parsing capabilities of the LFortran compiler, a research compiler currently in development. Using the Abstract Semantic Representation intermediate output of the compiler in this new work has made the translation simpler to accomplish. We also develop a map of the needed parallel constructs for a complete parallel language and begin to identify possible extensions to the existing Fortran language.

Fortran is the oldest high-level programming language that remains in use today and is one of the dominant languages used for compute-intensive scientific and engineering applications. However, Fortran has not kept up with the modern software development practices and tooling in the internet era. As a consequence, the Fortran developer experience has diminished. Specifically, lack of a rich general-purpose library ecosystem, modern tools for building and packaging Fortran libraries and applications, and online learning resources, has made it difficult for Fortran to attract and retain new users. To address this problem, an open source community has formed on GitHub in 2019 and began to work on the initial set of core tools: a standard library, a build system and package manager, and a communitycurated website for Fortran. In this paper we report on the progress to date and outline the next steps.

3 The role of Ristra at LANL post-ATDM 88Nodes

We discuss the p- and h p-versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schrödinger equation with a pseudo-potential term. As an interesting byproduct, we present for the first time in literature an explicit construction of the stabilization of the mass matrix. We present in detail the analysis of the p-version of the method, proving exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing h p-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the h p-spaces.

We present the results of the first Charged-Particle Transport Coefficient Code Comparison Workshop, which was held in Albuquerque, NM October 4-6, 2016. In this first workshop, scientists from eight institutions and four countries gathered to compare calculations of transport coefficients including thermal and electrical conduction, electron-ion coupling, inter-ion diffusion, ion viscosity, and charged particle stopping powers. Here, we give general background on Coulomb coupling and computational expense, review where some transport coefficients appear in hydrodynamic equations, and present the submitted data. Large variations are found when either the relevant Coulomb coupling parameter is large or computational expense causes difficulties. Understanding the general accuracy and uncertainty associated with such transport coefficients is important for quantifying errors in hydrodynamic simulations of inertial confinement fusion and high-energy density experiments.

Material equation-of-state (EOS) models, generally providing the pressure and internal energy for a given density and temperature, are required to close the equations of hydrodynamics. As a result they are an essential piece of physics used to simulate inertial confinement fusion (ICF) implosions. Historically, EOS models based on different physical/chemical pictures of matter have been developed for ICF relevant materials such as the deuterium (D-2) or deuterium-tritium (DT) fuel, as well as candidate ablator materials such as polystyrene (CH), glow-discharge polymer (GDP), beryllium (Be), carbon (C), and boron carbide (B4C). The accuracy of these EOS models can directly affect the reliability of ICF target design and understanding, as shock timing and material compressibility are essentially determined by what EOS models are used in ICF simulations. Systematic comparisons of current EOS models, benchmarking with experiments, not only help us to understand what the model differences are and why they occur, but also to identify the state-of-the-art EOS models for ICF target designers to use. For this purpose, the first Equation-of-State Workshop, supported by the US Department of Energy's ICF program, was held at the Laboratory for Laser Energetics (LLE), University of Rochester on 31 May-2nd June, 2017. This paper presents a detailed review on the findings from this workshop: (1) 5-10% model-model variations exist throughout the relevant parameter space, and can be much larger in regions where ionization and dissociation are occurring, (2) the D-2 EOS is particularly uncertain, with no single model able to match the available experimental data, and this drives similar uncertainties in the CH EOS, and ( 3) new experimental capabilities such as Hugoniot measurements around 100 Mbar and high-quality temperature measurements are essential to reducing EOS uncertainty.

error in the H1-norm. For the convection-dominated case, the streamline diffusion stabilization (aka SUPG) is also discussed.

We extend the conforming virtual element method (VEM) to the numerical resolution of eigenvalue problems with potential terms on a polytopic mesh. An important application is that of the Schrödinger equation with a pseudopotential term. This model is a fundamental element in the numerical resolution of more complex problems from the Density Functional Theory. The VEM is based on the construction of the discrete bilinear forms of the variational formulation through certain polynomial projection operators that are directly computable from the degrees of freedom. The method shows a great flexibility with respect to the meshes and provides a correct spectral approximation with optimal convergence rates. This point is discussed from both the theoretical and the numerical viewpoint. The performance of the method is numerically investigated by solving the quantum harmonic oscillator problem with the harmonic potential and a singular eigenvalue problem with zero potential for the first eigenvalues.

that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing hp-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the hp-spaces.

Electronic stopping power in warm dense matter can affect energy transport and heating in astrophysical processes and internal confinement fusion. For cold condensed matter systems, stopping power can be modeled from first-principles using real-time time-dependent density functional theory (DFT). However, high temperatures (10's to 100's of eV) may be computationally prohibitive for traditional Mermin-Kohn-Sham DFT. New experimental measurements in the warm dense regime motivates the development of first-principles approaches, which can reach these temperatures. We have developed a time-dependent orbital-free density functional theory, which includes a novel nonadiabatic and temperature-dependent kinetic energy density functional, for the simulation of stopping power at any temperature. The approach is nonlinear with respect to the projectile perturbation, includes all ions and electrons, and does not require a priori determination of screened interaction potentials. Our results compare favorably with Kohn-Sham for temperatures in the WDM regime, especially nearing 100 eV.

Electronic transport properties of warm dense matter, such as electrical or thermal conductivities and nonadiabatic stopping power, are of particular interest to geophysics, planetary science, astrophysics, and inertial confinement fusion (ICF). One example is the α-particle stopping power of dense deuterium-tritium (DT) plasmas, which must be precisely known for current small-margin ICF target designs to ignite. We have developed a time-dependent orbital-free density functional theory (TD-OF-DFT) method for ab initio investigations of the charged-particle stopping power of warm dense matter. Our current dependent TD-OF-DFT calculations have reproduced the recently well-characterized stopping power experiment in warm dense beryllium. For α-particle stopping in warm and solid-density DT plasmas, the ab initio TD-OF-DFT simulations show a lower stopping power up to ∼25% in comparison with three stopping-power models often used in the high-energy-density physics community.