Numerical PDEs/High-Order Discretization Modeling

We’re developing next-generation numerical methods to enable more accurate and efficient simulations of physical phenomena such as wave propagation, turbulent incompressible and high-speed reacting flows, shock hydrodynamics, fluid–structure interactions, and kinetic simulation. Our application-driven research is focused on designing, analyzing, and implementing new high-order finite difference, finite volume, and finite element discretization algorithms, with an emphasis on increased robustness, parallel scalability, and better utilization of modern computer architectures. View content related to Numerical PDEs/High-Order Discretization Modeling.

High-Order Finite Volume Methods

High-resolution finite volume methods are being developed for solving problems in complex phase space geometries, motivated by kinetic models of fusion plasmas. Techniques being investigated include conservative, high-order methods based on the method-of-lines for hyperbolic problems, as well as coupling to implicit solvers for fields equations. Mapped multiblock grids enable alignment of the grid coordinate directions to accommodate strong anisotrropy. The algorithms developed will be broadly applicable to systems of equations with conservative formulations in mapped geometries.

MFEM: Scalable Finite Element Discretization Library

Livermore’s open-source MFEM library enables application scientists to quickly prototype parallel physics application codes based on partial differential equations (PDEs) discretized with high-order finite elements. The MFEM library is designed to be lightweight, general and highly scalable, and conceptually can be viewed as a finite element toolkit that provides the building blocks for developing finite element algorithms in a manner similar to that of MATLAB for linear algebra methods. It has a number of unique features, including: support for arbitrary order finite element meshes and spaces with both conforming and nonconforming adaptive mesh refinement; advanced finite element spaces and discretizations, such as mixed methods, DG (discontinuous Galerkin), DPG (discontinuous Petrov-Galerkin) and Isogeometric Analysis (IGA) on NURBS (Non-Uniform Rational B-Splines) meshes; and native support for the high-performance Algebraic Multigrid (AMG) preconditioners from the HYPRE library.


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