Discontinuous Galerkin Methods

Discontinuous Galerkin (DG) methods approximate the solution by independent polynomials on each mesh element. The approximation space is broken:

\[ V_h = \{ v : v|_K \in \mathcal{P}_p(K) \quad \text{for each element } K \}. \]

For a state \(u\), Exasim stores element-local coefficients \(U_K\) such that

\[ u_h(x)|_K = \sum_j U_{Kj}\phi_j(x). \]

Neighboring elements do not share nodal degrees of freedom. Coupling occurs through numerical fluxes on faces.

Element-Local Weak Form

For a conservation law

\[ \frac{\partial u}{\partial t} + \nabla \cdot F(u) = S(u), \]

the DG weak form on an element \(K\) is obtained by multiplying by a test function \(\phi\), integrating by parts, and replacing the physical face flux with a numerical flux \(F^\ast\):

\[ \int_K \phi \frac{\partial u_h}{\partial t}\,dx - \int_K \nabla \phi \cdot F(u_h)\,dx + \int_{\partial K} \phi F^\ast \cdot n\,ds = \int_K \phi S(u_h)\,dx. \]

The numerical flux is responsible for stability, conservation across element interfaces, and boundary-condition enforcement.

Why DG Fits Exasim

DG methods match Exasim's code-generation and HPC design:

Property	Practical effect
Element-local polynomial data	Most kernels operate on compact element batches.
Local residual evaluation	Flux, source, EOS, AV, and visualization callbacks are pointwise or element-local.
High arithmetic intensity	High-order elements perform substantial work per loaded degree of freedom.
Flexible meshes	Curved, unstructured, and boundary-fitted meshes fit naturally.
Conservative face coupling	Interface fluxes preserve conservation when model fluxes are consistent.
GPU suitability	Element and quadrature loops map to many independent GPU work items.

LDG And HDG

Exasim uses two DG solver paths:

• LDG keeps element solution unknowns as the primary global state and uses matrix-free Newton-GMRES.
• HDG introduces trace unknowns on the mesh skeleton, statically condenses element unknowns, and solves a matrix-based global trace problem.

Both paths use the same model callbacks for fluxes, sources, boundary terms, initial conditions, EOS, artificial viscosity, QoI, and visualization fields.

Conservation And Consistency

DG conservation depends on using compatible interior and boundary fluxes. For an interior face shared by elements \(K^-\) and \(K^+\), the numerical flux must be single-valued on the face. Conservation follows because the same face flux enters the two neighboring residuals with opposite normal directions.

Consistency means that when the traces from both sides match the exact solution, \(F^\ast\) reduces to the physical flux. In Exasim, this consistency is encoded in the user/model boundary and interface callbacks.

Practical Guidance

Choose DG/LDG/HDG settings together with the PDE model:

Need	Recommended direction
Lowest setup complexity and matrix-free iterations	LDG
Diffusion-dominated HDG formulations with fewer global unknowns	HDG
Large implicit simulations where trace systems are cheaper than full state systems	HDG
Problems where matrix-free residual products are preferred over assembled matrices	LDG
GPU-heavy runs with high-order local kernels	Either LDG or HDG; benchmark with the target preconditioner

See also Physics Models for choosing ModelC, ModelD, or ModelW.