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Nonlinear Solvers

Exasim solves nonlinear algebraic systems with Newton iteration. Nonlinearity can come from fluxes, sources, EOS closures, boundary conditions, artificial viscosity, auxiliary equations, or implicit time integration.

Newton Formulation

Let \(U\) denote the unknown vector for the active solver path:

  • LDG: element state unknowns;
  • HDG: trace unknowns after static condensation.

The nonlinear residual is

\[ R(U)=0. \]

At Newton iteration \(k\), the correction \(\Delta U^k\) satisfies

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

where

\[ J(U^k) = \frac{\partial R}{\partial U}(U^k). \]

The state is updated as

\[ U^{k+1} = U^k + \alpha \Delta U^k, \]

where \(\alpha\) may be reduced by the solver when damping is required.

Sources Of Nonlinearity

Source Example
Convective fluxes Euler and Navier-Stokes equations.
Diffusive fluxes Temperature- or state-dependent viscosity.
Source terms Chemistry, relaxation, body forces.
EOS and material laws Pressure, temperature, transport coefficients.
Boundary fluxes Wall, inflow, outflow, and Riemann boundary states.
DIRK stages Implicit stage residuals for time-dependent simulations.

LDG Versus HDG Newton Systems

Solver path Newton unknown Jacobian action
LDG Element state vector Matrix-free residual-based matrix-vector products.
HDG Trace vector after condensation Matrix-based trace operator assembled from element Jacobians.

The nonlinear convergence criteria are controlled by fields such as NewtonIter, NewtonTol, NLiter, and NLtol, depending on frontend or text input path.

Convergence Diagnostics

Solver output typically reports the solution norm, residual norm, GMRES iterations, and Newton update quality. Use these together:

  • residual norm decreasing but GMRES stagnating suggests a linear solver or preconditioner issue;
  • GMRES convergence with poor Newton progress suggests nonlinear stiffness, bad initial conditions, or inconsistent model Jacobians;
  • sudden residual growth often indicates invalid physics states, boundary condition errors, or excessive time step size.

Practical Guidance

  • Start with conservative nonlinear tolerances when validating a new model.
  • Verify boundary conditions and flux Jacobians before tuning GMRES.
  • For stiff parameter sweeps, use warm starts only when neighboring parameter points are physically close.
  • For time-dependent runs, distinguish a failed DIRK stage from a failed steady-state nonlinear solve; the remedies may differ.