atl_scheme_module Module

The scheme describes the discretization to use in the simulation.

There are two parts that need to be configured:

• The spatial discretization
• And the temporal discretization

Optionally a stabilization method may be defined for the scheme, see the atl_stabilization_module for more details on that definition.

The kind of spatial discretization is chosen via the name setting within the spatial table. The following spatial schemes are available:

• 'modg' Modal DG discretization in 3D
• 'modg_2d' Modal DG disretization in 2D
• 'modg_1d' Modal DG discretization in 1D

The main configuration option for the spatial discretization is the polynomial degree to use to represent the state in the DG elements. This polynomial degree is set by the option m in the spatial table. It may either be a simple scalar value, defining a single polynomial degree to be used for all elements in the domain, or a table that provides polynomial degrees based on the local refinement level of elements. This can be achieved either by providing the polynomial degree for each level individually or by choosing a predefined scheme to choose the polynomial degree for the levels.

Individual definitions take the following form:

  m = {
    { level = 4, m = 4 },
    { level = 5, m = 6 }
  }

A predefined scheme is offered by 'fixedfact', where the polynomial degree on each level is computed by the following formula.

  m(iLevel) = nint( base_order * factor**(maxLevel-iLevel)) - 1

Here the base_order and factor need to be defined in the configuration, where the base_order sets the minimal polynomial degree (+1) that is to be used on the finest level. factor defines the factor, by which the scheme order is to be increased by each level. The polynomial degree definition for this case looks as follows:

  m = {
    predefined = 'fixedfact',
    base_order = 4,
    factor     = 1.5
  }

The default factor for the 'fixedfact' scheme is $\sqrt{2}$, which allows for approximately the same time step restriction across the levels for hyperbolic equations.

Besides the polynomial degree m it is also possible to choose the polynomial space to use for multidimensional representation. This modg_space is either 'P' or 'Q'. P indicates a multidimensional polynomial, where the sum of the mode indices is at most equal to the configured polynomial degree m. Q indicates that each index in the different dimensions itself may at most be m. The 'Q' space is the default but requires more computational effort and memory especially for 3D simulations.

The explicit time integration is configured by the temporal table within scheme. Following schemes are available:

• 'explicitEuler', only for testing! (unstable)
• 'explicitRungeKutta', with steps=4
• 'imexRungeKutta', with steps=4
• 'explicitRungeKuttaTaylor', with arbitrary number of steps
• 'explicitSSPRungeKutta', with steps=2

The 'imexRungeKutta' scheme should be used when penalization terms are to be used in the flow simulations. The 'explicitRungeKuttaTaylor' is mainly intended for the solution of linear equation systems.

The time step width is controlled by a control subtable and the time step can either be chosend adaptively according to the CFL condition, or set as a fixed time step. See also atl_global_time_integration_module.

A complete definition of the scheme without the optional stabilization table takes the following form:

  scheme = {
    spatial = {
      name = 'modg',
      modg_space = 'Q',
      m = 11
    },
    temporal = {
      name = 'explicitRungeKutta',
      steps = 4,
      control = {
        name = 'cfl',
        cfl = 0.8
      }
    }
  }

For details on the optional stabilization see the atl_stabilization_module.