The time derivatives appearing in the Coefficient Form and General Form PDE templates are shown in the user interface and documentation as partial derivatives of the dependent variable with respect to time. This is true only as long as the mesh on which the dependent variables are represented is fixed. When using Moving Mesh or Deformed Geometry functionality, the mesh nodes and elements may be moving relative to the frame chosen under Discretization in the interface settings. Time derivatives used in domain PDEs (that is, on the full dimension of the geometry) are then interpreted as a total derivative with respect to time, evaluated for fixed coordinates in the chosen frame.

The dependent variables can be seen as functions of mesh coordinates Xm and time t. If the mesh is moving, then Xm(t) is also a function of time when seen from a point x fixed in space. Therefore

and similarly for second-order and mixed derivatives. For PDEs on lower dimensions, for example a Coefficient Form Boundary PDE, this relation cannot be used since it is in general not possible to hold the position x fixed in space if the mesh motion is not purely tangential. For this reason, time derivatives in PDEs on lower dimensions are defined simply as the mesh frame derivative:

The equation-based interfaces have an Equation form setting that controls the interpretation of time derivatives, depending on whether the equation is seen as a time-domain equation, a frequency domain equation using the harmonic ansatz

where ω = 2πf is the angular frequency, or an eigenvalue equation using

When the equation form is set to Time domain, then the eigenvalue ansatz is used when solving using an Eigenvalue or Eigenfrequency study step. For all other solvers, the time derivative variable ut evaluates based on the time derivative of the degrees of freedom, which are zero in the stationary solver used by both Stationary and Frequency Domain study steps.

When the equation form is set to Frequency domain, then the harmonic ansatz is always used, meaning that time derivative variables are evaluated as ut = iωu for all solvers and study step types. But the angular frequency ω is still by default defined by the study step. A Frequency Domain study step sets the angular frequency to ω = 2πf where the frequency f is given in the study step. An Eigenvalue or Eigenfrequency study step defines ω = iλ, turning the frequency domain problem into a corresponding eigenvalue problem. It is also possible to explicitly set the frequency in the PDE interface settings.

The default equation form setting when working from the user interface is Study controlled, which means that the frequency-domain interpretation of time derivatives will be used in Frequency Domain study steps with the frequency supplied by the study step. For all other study types, the time-domain interpretation will be used.