Theory

This section explains the numerical methods and solver technology behind Exasim. It connects the mathematical PDE model to the generated C++ backend: spatial discretization, time integration, nonlinear solution, linear Krylov solution, preconditioning, MPI parallelism, and GPU execution.

The requested source document Exasim.tex was not present in the repository or attached files when this section was revised. The content here is therefore grounded in the current Exasim implementation, the existing internal LDG notes, and the public configuration fields documented in the reference pages.

Numerical Pipeline

Exasim transforms a user PDE model into a sequence of algebraic problems:

flowchart LR
  PDE["PDE model<br/>u, q, w, v, fluxes, sources"] --> DG["DG spatial discretization"]
  DG --> TIME["Steady / DIRK / BDF / ERK time discretization"]
  TIME --> NEWTON["Newton nonlinear iteration"]
  NEWTON --> GMRES["GMRES Krylov solve"]
  GMRES --> PREC["Preconditioner"]
  PREC --> UPDATE["Solution update"]

The main choices are:

Layer	Exasim concept	Main user-facing controls
PDE model	ModelC, ModelD, ModelW, callbacks, generated kernels	model, modelfile, physicsparam
Spatial discretization	LDG or HDG-style DG discretization	discretization, hybrid, spatialScheme
Time integration	steady solve, DIRK, BDF, ERK selectors	tdep, temporalscheme, torder, nstage, dt
Nonlinear solve	Newton iteration	NewtonIter, NewtonTol, NLiter, NLtol
Linear solve	GMRES and related Krylov controls	GMRESiter, GMRESrestart, GMREStol
Preconditioning	local, Schwarz, polynomial, reduced-basis controls	preconditioner, ppdegree, RBdim
Parallelism	MPI domain decomposition	mpiprocs, generated dmd, partition files
GPU execution	CUDA/HIP/Kokkos backend variants	build options and installed package target

How To Use This Section

Page	Purpose
DG methods	Common DG ideas: broken polynomial spaces, numerical fluxes, conservation, and high-order accuracy.
LDG	LDG formulation used by Exasim, including matrix-free Newton-GMRES.
HDG	HDG formulation, trace variables, static condensation, and matrix-based trace solves.
Temporal discretization	Steady-state and time-dependent workflows, with emphasis on DIRK stage equations.
Nonlinear solvers	Newton residuals, Jacobians, updates, and convergence controls.
Linear solvers	GMRES, restarted Krylov methods, and LDG/HDG matrix-vector products.
Preconditioning	Implemented preconditioner families and practical selection guidance.
Parallel computing	MPI decomposition, communication patterns, and distributed-memory implications.
GPU computing	CUDA/HIP/Kokkos execution model and memory-movement considerations.
Scalability	Strong/weak scaling, arithmetic intensity, memory footprint, and practical tradeoffs.
Algorithmic flow	End-to-end HDG and LDG solve workflows.
LDG implementation deep dive	Detailed LDG weak form and residual construction.
Block-diagonal Jacobian	Exact element-block Jacobian of the LDG residual path.
References	Background literature and implementation references.

Relationship To Other Documentation

• Use Physics Models to decide how to represent a PDE system in terms of u, q, w, v, EOS, AV, and coupling terms.
• Use Frontend Configuration and pdeapp.txt fields to set solver options.
• Use Application Modes to choose a built-in, shared-library, frontend, parameter-sweep, or postprocessing workflow.
• Use Internals when debugging backend object construction, generated files, and runtime data flow.