Tempus: Add Ability to Integrate Auxiliary Variables

[ccober6]

@trilinos/tempus

Expectations

Have the ability to time integrate auxiliary variables.

There is a difference between the PhysicsState, p, and an auxiliary variable, y. The PhysicState is physics data that does not need time integration. It is just data at each time step that is needed to evaluate f(x,xDot,t,p). An auxiliary variable, y, however does need time integration, but has been separated from the solution variable, x, because it is a simple update or has a different time-integration.

To start with we will assume that the solution, x, and the auxiliary variables, y, are integrated with the same Stepper, but is a simple update from the SolutionState (e.g., _x{n-1}, _x__{n}, _xDot{n-1}, _xDot{n}, _y__{n-1}, _yDot{n-1}, and RK stage values) and other values available from the TimeDerivative class.

Motivation and Context

The ability to segregate the solution, x, and auxiliary variables, y, can be a substantial computational savings because of the reduction in size of the solution vector from {x, y} to just {x} and simple update of {y}.

So, there two types of variables we would like to handle in Tempus, solution variables (implemented) and auxiliary variables (not implemented yet, but has been a requirement since the beginning).

• Solution Variables, x.
  • Explicit time steppers and time derivatives, xDot
    • Tempus relies on the application through the ModelEvaluator to provide xDot with xDot = f(x,t).
    • There is not a way for Tempus to generate xDot without x^n and x^{n-1} or x^n without x^{n-1} and xDot (chicken-n-egg problem).
    • However, one could do an extrapolation from just past values, e.g., x^{n-1} and x^{n-2}.
  • Implicit time steppers and time derivatives, xDot
    • Tempus evaluates an xDot that is consistent with time stepper and passes it to the ModelEvaluator so it can evaluate f(x, xDot, t) = 0.
    • Similar chicken-n-egg problem but iterative solver makes a guess that allows to create an estimated xDot.
  • IMEX time steppers and time derivatives, xDot
    • At the end of the stage, there is a way to determine the time derivative (not implemented but easy to add; note this is not the pseudo time derivative but stageXDot).
• Auxiliary Variables, y. (not implemented yet)
  • If you have yDot = g(x,xDot,y,t) (for explicit) or g(x,xDot,y,yDot,t) = 0 (for implicit), I would suggest making it part of the solution variables, x.
  • Otherwise, ...
    • Compute an extrapolated yDot from x and y values, e.g., x^n, x^{n-1}, y^n and y^{n-1}.
    • Use a chain rule representation to calculate yDot from xDot.
  • Also could time step the auxiliary variables with a simpler/cheaper time stepper.
    • You have yDot = g(x,xDot,y,t) or g(x,xDot,y,yDot,t) = 0 just pass it to another time stepper.

Definition of Done

• [x] Add auxiliary variables, y, to SolutionState.
• [ ] Incorporate y to TimeDerivative class.
• [x] Develop means for application to provide update function to Stepper. (This may be a prototype for a new ModelEvaluator).
• [ ] Have TempusWrapperModelEvaluator use y and compute yDot.
• [ ] Either implement the new auxiliary variables in every Stepper, or have the Stepper error out if auxiliary variables are used.
• [ ] Add test to cover this new capability.

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