Local Discontinuous Galerkin¶

Local Discontinuous Galerkin (LDG) rewrites PDEs with gradients as first-order systems by introducing an auxiliary gradient-like variable \(q\). With the sign convention used in the Physics Models documentation, a diffusion-type model is

\[ q + \nabla u = 0, \]

\[ \frac{\partial u}{\partial t} + \nabla \cdot F(u,q,w,v,x,t;\mu) = S(u,q,w,v,x,t;\mu). \]

Some internal implementation notes use the opposite sign convention \(q-\nabla u=0\). The sign is a convention as long as the flux and source definitions are consistent with the generated model.

Why LDG Is Useful¶

LDG keeps the discretization close to the original DG structure:

\(q\) is recovered from local element and face data.
The \(u\) residual is assembled from volume fluxes, sources, and face terms.
Matrix-vector products can be computed without forming the full Jacobian.
The implementation is well suited to generated residual kernels and GPU execution.

Exasim LDG Residual Path¶

The detailed residual construction is documented in LDG implementation deep dive. At a high level, an LDG residual evaluation performs:

flowchart TD
  U["element state u"] --> UHAT["face state / numerical trace"]
  UHAT --> Q["recover auxiliary q"]
  Q --> W["evaluate auxiliary w, if present"]
  W --> AV["evaluate artificial viscosity, if enabled"]
  AV --> VOL["volume residual: flux + source"]
  VOL --> FACE["face residual: boundary/interface flux"]
  FACE --> R["LDG residual R(u)"]

The model contract supplies pointwise terms such as flux, source, fbou, ubou, initu, initq, and optional EOS/AV/visualization callbacks.

Matrix-Free Newton-GMRES¶

For nonlinear LDG problems, Exasim solves

\[ R(U)=0 \]

by Newton iteration. The linearized Newton correction satisfies

\[ J(U^k)\Delta U = -R(U^k). \]

In the LDG path, GMRES uses matrix-vector products

\[ J(U)v \approx \frac{R(U+\epsilon v)-R(U)}{\epsilon}, \]

or derivative-aware paths when enabled by the build. This avoids storing the full global Jacobian. The user-facing controls include matvecorder, matvectol, GMRESrestart, GMRESiter, GMREStol, preconditioner, ppdegree, and RBdim.

Block-Diagonal Jacobian¶

Exasim includes an implementation-level derivation of the exact element-block LDG Jacobian. This is useful for understanding local preconditioning and the chain rule through the LDG residual pipeline. See Block-diagonal Jacobian.

Advantages And Limitations¶

Aspect	LDG behavior
Global unknowns	Element-local state unknowns are globally iterated.
Matrix storage	Full global Jacobian need not be stored for matrix-free GMRES.
Memory footprint	Lower matrix memory than assembled methods, but Krylov vectors still scale with state DOFs.
Preconditioning	Local and Schwarz-style preconditioners are important for robustness.
GPU use	Residual and matrix-vector products are dominated by element and face kernels.
Solver iterations	May require more Krylov iterations than a well-preconditioned assembled trace system.

When To Choose LDG¶

Choose LDG when:

you want matrix-free implicit solves;
the full state system is manageable;
residual evaluation is cheap relative to matrix assembly/storage;
the target backend benefits from repeated local operator evaluations;
you are developing or debugging a new PDE model and want a direct DG path.

For diffusion-dominated problems with many element interior unknowns, compare against HDG, which reduces the globally coupled system to trace unknowns.