Operator Splitting

An operator splitting scheme is used to solve e.g. a reaction diffusion equation.

(2)\[u_t - Δu = R(u,y)\]

The equation is solved with only reaction and diffusion part separately.

Fig. 108 Scheme of Godunov splitting for (2)

Fig. 109 Scheme of Strang splitting for (2)

GodunovSplitting

The Godunov operator splitting, as shown in Fig. 108 is first order consistent.

C++ code:

OperatorSplitting::Godunov<
  /*first timestepping scheme*/,
  /*second timestepping scheme*/
>

Python settings:

"GodunovSplitting": {
  "numberTimeSteps": 1,
  "timeStepWidth": 1e-1,
  "endTime": 10.0,
  "logTimeStepWidthAsKey": "dt_3D",
  "durationLogKey": "duration_total",
  "timeStepOutputInterval": 10,

  "connectedSlotsTerm1To2": [0],  # list of slots of term 2 that are connected to the slots of term 1
  "connectedSlotsTerm2To1": [0],  # list of slots of term 1 that are connected to the slots of term 2

  "Term1": {
    "ExplicitEuler" : {  # these are the properties for the first timestepping
    }
  },
  "Term2": {
    "ImplicitEuler" : {  # these are the properties for the second timestepping
    }
  }
}

As the GodunovSplitting is also a time stepping scheme, the common options described under TimeSteppingSchemesOde are also in use here.

The GodunovSplitting scheme perform two data transfers:

• At the beginning from timestepping 1 to timestepping 2

• at the end from timestepping 2 to timestepping 1

The Coupling class is the same as GodunovSplitting, except that the keyword is “Coupling” instead of “GodunovSplitting”.

StrangSplitting

The Strang operator splitting, as shown in Fig. 109 is first order consistent.

C++ code:

OperatorSplitting::Strang<
  /*first timestepping scheme*/,
  /*second timestepping scheme*/
>

Python settings:

"StrangSplitting": {
  "numberTimeSteps": 1,
  "timeStepWidth": 1e-1,
  "endTime": 10.0,
  "logTimeStepWidthAsKey": "dt_3D",
  "durationLogKey": "duration_total",
  "timeStepOutputInterval": 10,

  "connectedSlotsTerm1To2": [0],  # list of slots of term 2 that are connected to the slots of term 1
  "connectedSlotsTerm2To1": [0],  # list of slots of term 1 that are connected to the slots of term 2

  "Term1": {
    "ExplicitEuler" : {  # these are the properties for the first timestepping
    }
  },
  "Term2": {
    "ImplicitEuler" : {  # these are the properties for the second timestepping
    }
  }
}

As the StrangSplitting is also a time stepping scheme, the common options described under TimeSteppingSchemesOde are also in use here.

The StrangSplitting scheme perform two data transfers: * At the beginning from timestepping 1 to timestepping 2 * at the end from timestepping 2 to timestepping 1

Which values to transfer can be specified by the settings connectedSlotsTerm1To2 and connectedSlotsTerm2To1. See the notes on Connector Slots.