Solving Time-Dependent Problems

The general form equation shown in Equation 16-1, as well as the coefficient form equation in Equation 16-2, contain time-derivative terms of the same form. These terms only take effect for Time Dependent, Eigenvalue, and Eigenfrequency study steps, and derived versions of these. When solving a Stationary, Frequency Domain, or similar study step, the solvers assume that all time derivatives are zero, so the values of the ea and da coefficients do not matter.


	To activate the da and ea coefficients and convert the model into a time-dependent model, select a Time Dependent study.

When solving a Time Dependent study step, the mass coefficient, ea, becomes important. The name mass coefficient, or mass matrix in case of a system of equations, stems from the fact that in many physics applications, ea contains the mass density. The da coefficient in such equations usually represents damping of wave-like phenomena. However, if ea = 0, then da is often called the mass coefficient instead. The default settings are ea = 0 and da = 1, representing a parabolic time-dependent PDE such as the heat equation. Using ea = 1 and da = 0 represents an undamped wave equation.


	When solving a Time Dependent study step, the time variable is called t and can be used anywhere in equation coefficients. For other study steps, t is undefined. If you want to solve a model that depends explicitly on time using a Stationary study, you must first define a model parameter called t and give it a suitable value.

If, for a system of equations, the ea matrix is nonzero and singular, or if ea = 0 and da is singular, the system becomes a differential-algebraic equation (DAE) system. The COMSOL Multiphysics solvers for time-dependent problems handle DAEs.


	Time-Dependent Solver

Using Mixed Space-Time Derivatives

The coefficient forms in equation Equation 16-2 only contain coefficients for pure space and time derivatives up to second order. The only directly available time-derivative coefficients are therefore ea and da, using the subscript a because they are similar to the a coefficient in the absorption term, except that they multiply ∂2u/∂t2 and ∂u/∂t instead of u. In analogy, it is possible to define coefficients ec, eα, eβ and dc, dα, dβ for mixed space-time derivatives, such that the equation becomes instead

These mixed coefficients are not directly available in the general or coefficient form PDE models. Instead, enter them in the existing γ and f terms:


	In 1D, add -d_cuxt-d_alut to the γ term, and add -d_be*uxt to the f term, and similarly for second-order derivatives.


	In 2D, add -d_cuxt-d_al1ut to the first γ component, and add -d_cuyt-d_al2ut to the second γ component. Add -d_be1uxt-d_be2uyt to the f term, and similarly for second-order derivatives.

Using Time Derivatives in Boundary Conditions

To specify a flux or source boundary condition containing time-derivative terms as in

simply add the terms -e_q*utt-d_q*ut to the g term, and provide appropriate values or expressions for the coefficients eq and dq in, for example, a Global Equations Settings window.


	Constraints and Dirichlet boundary conditions must not contain time derivatives like ut and utt in the R and r coefficients unless they are enforced weakly, using weak constraints. See Boundary Conditions.