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Time Integration

After discretization in configuration and velocity space, GUERNICA advances the semi-discrete system

\[ \mathbf{M} \frac{d \mathbf{f}}{dt} = \mathbf{R}_{\mathrm{transport}} + \mathbf{R}_{\mathrm{collisions}} + \mathbf{R}_{\mathrm{sources}}, \]

where \(\mathbf{M}\) is the DG mass matrix and \(\mathbf{R}\) is the sum of transport, collision, and source contributions.

This section describes how the resulting system of ordinary differential equations is integrated in time.

Explicit time integration

In its current formulation, GUERNICA employs explicit time integration for all operators.

The right-hand side is evaluated as

\[ \frac{d \mathbf{f}}{dt} = \mathbf{M}^{-1} \left( \mathbf{R}_{\mathrm{transport}} + \mathbf{R}_{\mathrm{collisions}} + \mathbf{R}_{\mathrm{sources}} \right), \]

with the inverse mass applied via an iterative solve.

Explicit methods are favored for their simplicity and their compatibility with matrix-free operator evaluation.

Time-stepping scheme

GUERNICA uses a strong-stability-preserving Runge–Kutta (SSP-RK) scheme for time advancement. In particular, a third-order SSP Runge–Kutta method is employed in most simulations.

This choice provides:

  • third-order accuracy in time,
  • good nonlinear stability properties for hyperbolic transport,
  • straightforward coupling with collision operators.

Each Runge–Kutta stage requires a full evaluation of the right-hand side and an inverse mass application.

Stability constraints

The time step \(\Delta t\) is constrained by both transport and collision processes.

Transport (CFL) constraint

The DG transport operator imposes a CFL condition of the form

\[ \Delta t \le C_{\mathrm{CFL}} \frac{h}{|\mathbf{v}_j| (2p+1)}, \]

where: - \(h\) is a characteristic element size, - \(p\) is the DG polynomial degree, - \(|\mathbf{v}_j|\) is the magnitude of the discrete velocity, - \(C_{\mathrm{CFL}}\) is a method-dependent constant.

Because the CFL condition depends on the maximum discrete velocity, velocity-space truncation directly influences the allowable time step.

Collision stiffness

Collision operators introduce additional stiffness:

  • Ionization imposes a constraint proportional to \(\nu_{\mathrm{iz}}^{-1}\).
  • Charge exchange may introduce stiffness in dense, cold regions.
  • BGK relaxation imposes a constraint proportional to \(\nu_{\mathrm{nn}}^{-1}\).

In regimes where collision frequencies dominate transport, explicit time stepping may require prohibitively small time steps.

Implementation in GUERNICA

Time integration and RHS assembly

  • SumTDep.hxx / SumTDep.cxx
  • guernica.cxx

The time integrator advances the sum of transport, collision, and source operators, with inverse mass application performed via an iterative solve.