Co-Design: Deploying Leading-Edge Computing Capabilities



Laghos (LAGrangian High-Order Solver) is a miniapp that solves the time-dependent Euler equations of compressible gas dynamics in a moving Lagrangian frame using unstructured high-order finite element spatial discretization and explicit high-order time-stepping.

Laghos is based on the discretization method described in the following article:

V. Dobrev, Tz. Kolev and R. Rieben
High-order curvilinear finite element methods for Lagrangian hydrodynamics
SIAM Journal on Scientific Computing, (34) 2012, pp. B606–B641.

Laghos captures the basic structure of many other compressible shock hydrocodes, including the BLAST code at Lawrence Livermore National Laboratory. The miniapp is built on top of a general discretization library, MFEM, thus separating the pointwise physics from finite element and meshing concerns.

In each time step, the problem is ultimately formulated as solving a big system of ordinary differential equations (ODE) for the unknown (high-order) velocity, internal energy and mesh nodes (position). The left-hand side of this ODE is controlled by mass matrices (one for velocity and one for energy), while the right-hand side is constructed from a force matrix.

Laghos supports two options for deriving and solving the ODE system, namely the full assembly and the partial assembly methods. Partial assembly is the main algorithm of interest for high orders. For low orders (e.g. 2nd order in 3D), both algorithms are of interest.

The full assembly option relies on constructing and utilizing global mass and force matrices stored in compressed sparse row (CSR) format.

The partial assembly option (also known as sum factorization) defines only the local action of those matrices, which is then used to perform all necessary operations. As the local action is defined by utilizing the tensor structure of the finite element spaces, the amount of data storage, memory transfers, and FLOPs are lower (especially for higher orders).

Other computational motives in Laghos include the following:

  • Support for unstructured meshes, in 2D and 3D, with quadrilateral and hexahedral elements (triangular and tetrahedral elements can also be used, but with the less efficient full assembly option). Serial and parallel mesh refinement options can be set via a command-line flag.
  • Explicit time-stepping loop with a variety of time integrator options. Laghos supports Runge-Kutta ODE solvers of orders 1, 2, 3, 4 and 6.
  • Continuous and discontinuous high-order finite element discretization spaces of runtime-specified order.
  • Moving (high-order) meshes.
  • Separation between the assembly and the quadrature point-based computations.
  • Point-wise definition of mesh size, time-step estimate and artificial viscosity coefficient.
  • Constant-in-time velocity mass operator that is inverted iteratively on each time step. This is an example of an operator that is prepared once (fully or partially assembled), but is applied many times. The application cost is dominant for this operator.
  • Time-dependent force matrix that is prepared every time step (fully or partially assembled) and is applied just twice per “assembly”. Both the preparation and the application costs are important for this operator.
  • Domain-decomposed MPI parallelism.
  • Optional in-situ visualization with GLVis and data output for visualization / data analysis with VisIt.

Additional information can be found in the Laghos GitHub page,
The Laghos miniapp is part of the CEED software suite, a collection of software benchmarks, miniapps, libraries and APIs for efficient exascale discretizations based on high-order finite element and spectral element methods. See for more information and source code availability.
The CEED research is supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of two U.S. Department of Energy organizations (Office of Science and the National Nuclear Security Administration) responsible for the planning and preparation of a capable exascale ecosystem, including software, applications, hardware, advanced system engineering and early testbed platforms, in support of the nation’s exascale computing imperative.

Software Download

Download Laghos (LLNL-CODE-734707)


You can reach the Laghos team by emailing or by leaving a comment in the issue tracker.