Overview of the structure of Trixi

Trixi is designed as a library of components for discretizations of hyperbolic conservation laws. Thus, it is not a monolithic PDE solver that is configured at runtime via parameter files, as it is often found in classical numerical simulation codes. Instead, each simulation is configured by pure Julia code. Many examples of such simulation setups, called elixirs in Trixi, are provided in the examples folder.

Trixi uses the method of lines, i.e., the full space-time discretization is separated into two steps; the spatial semidiscretization is performed at first and the resulting ODE system is solved numerically using a suitable time integration method. Thus, the main ingredients of an elixir designed to solve a PDE numerically are the spatial semidiscretization and the time integration scheme.

Semidiscretizations

Semidiscretizations are high-level descriptions of spatial discretizations specialized for certain PDEs. Trixi's main focus is on hyperbolic conservation laws represented in a SemidiscretizationHyperbolic. Such semidiscretizations are usually named semi in Trixi.

semidiscretization_overview

The basic building blocks of a semidiscretization are

  • a mesh describing the geometry of the domain
  • a set of equations describing the physical model
  • a solver describing the numerical approach

In addition, a semidiscretization bundles initial and boundary conditions, and possible source terms. These different ingredients of a semidiscretization can be configured individually and combined together. When a semidiscretization is constructed, it will create an internal cache, i.e., a collection of setup-specific data structures, that is usually passed to all lower level functions.

Due to Trixi's modular nature using Julia's multiple dispatch features, new ingredients can be created specifically for a certain combination of other ingredients. For example, a new mesh type can be created and implemented at first only for a specific solver. Thus, there is no need to consider all possible combinations of meshes, equations, and solvers when implementing new features. This allows rapid prototyping of new ideas and is one of the main design goals behind Trixi. Below is a brief overview of the availability of different features on different mesh types.

FeatureTreeMeshStructuredMeshUnstructuredMesh2DP4estMeshDGMultiMeshFurther reading
Spatial dimension1D, 2D, 3D1D, 2D, 3D2D2D, 3D1D, 2D, 3D
CoordinatesCartesiancurvilinearcurvilinearcurvilinearaffine
Connectivityh-nonconformingconformingconformingh-nonconformingconforming
Element typeline, square, cubeline, quadᵃ, hexᵃquadᵃquadᵃ, hexᵃsimplex, quadᵃ, hexᵃ
Adaptive mesh refinementAMRCallback
Solver typeDGSEMDGSEMDGSEMDGSEMDGMulti
Domainhypercubemapped hypercubearbitraryarbitraryarbitraryᵇ
Weak formVolumeIntegralWeakForm
Flux differencingVolumeIntegralFluxDifferencing
Shock capturingVolumeIntegralShockCapturingHG
Nonconservative equationse.g., GLM MHD or shallow water equations

ᵃ: quad = quadrilateral, hex = hexahedron ᵇ: curved meshes supported for SBP and GaussSBP approximation types for VolumeIntegralFluxDifferencing solvers on quadrilateral and hexahedral DGMultiMeshes (non-conservative terms not yet supported)

Time integration methods

Trixi is compatible with the SciML ecosystem for ordinary differential equations. In particular, a spatial semidiscretization can be wrapped in an ODE problem using semidiscretize, which returns an ODEProblem. This ODEProblem is a wrapper of Trixi.rhs!(du_ode, u_ode, semi, t), which gets called in ODE solvers. Further information can be found in the section on time integration methods.

Next steps

We explicitly encourage people interested in Trixi to have a look at the examples bundled with Trixi to get an impression of what is possible and the general look and feel of Trixi. Before doing that, it is usually good to get an idea of how to visualize numerical results.

If you like learning by doing, looking at the tutorials and trying to mix your own elixirs based thereon is probably a good next step. Otherwise, you can further dig into the documentation by looking at Trixi's basic building blocks.