Modelling

DAEs

DAEs are a system of differential equations in one independent variable. In DAEs, the derivatives of some dependent variables might not explicitly appear.

The system of DAES can be converted to a system of ODEs (where the derivatives are explicit) by differentiating the equations with respect to the independent variable. The number of times they need to be differentiated to convert to a system of ODEs provides the index. To solve a system of DAEs they are typically first reduced to an index of 1 (often more efficient than fully converting to ODE and solving), although techniques for solving high index DAEs also exist. For more information see wolfram or wikipedia.

The primary solver used by OpenModelica for solving index 1 or less DAEs is DASSL. Some information on the solver is provided here. Another popular DAE solver is IDA (which is derived from DASPK).

PDEs

OpenModelica can only solve DAEs or ODEs, so one of a number of different approximation methods will be required for any PDEs:

• Finite difference method
• Finite element method
• Finite volume method
• Weak form

Discrete Events

Have to make sure the number of equations is equal to the number of variables at all times.

Considerations

From Modelica.Fluid documentation:

“The resulting simulation performance however often strongly depends on the model structure and modeling assumptions made. In particular the direct connection of fluid volumes generally results in high-index DAEs for the pressures. The direct connection of flow models generally results in systems of implicit nonlinear algebraic equations.”