Time Stepping¶
The semidiscrete system¶
A transient model adds a time-derivative term to its weak form. Discretizing in space only, and leaving time continuous, turns the problem into a system of ordinary differential equations for the time-dependent degrees of freedom \(\mathbf{x}(t)\):
| Symbol | Meaning |
|---|---|
| \(M\) | Mass matrix, assembled from the time-derivative term (for example the \(\sigma\,\partial_t\) term of an eddy-current problem or the \(\rho c_p\,\partial_t\) term of transient heat conduction) |
| \(A\) | System (stiffness) matrix from the spatial part of the weak form |
| \(\dot{\mathbf{x}}\) | Time derivative of the degrees of freedom |
| \(\mathbf{b}(t)\) | Time-dependent right-hand side |
This is the method of lines: space is discretized as usual, and what remains is to advance \(\mathbf{x}(t)\) in time.
Implicit time integration¶
The unsteady runner advances the solution in discrete time steps of size \(\Delta t\), from \(\mathbf{x}^n\) at time \(t^n\) to \(\mathbf{x}^{n+1}\). μfem uses implicit integration, which stays stable even for the stiff, diffusive systems typical of eddy-current and heat-conduction problems, where explicit schemes would demand impractically small steps.
μfem advances with the first-order backward Euler scheme, which approximates the time derivative with the new (unknown) state from the single stored previous step,
Substituting into the semidiscrete system and collecting the unknown \(\mathbf{x}^{n+1}\) gives
Every time step is therefore an ordinary linear system of the same form as a stationary solve: an effective matrix \(\frac{1}{\Delta t} M + A\) on the left, and a right-hand side that carries the previous step \(\mathbf{x}^n\). It is assembled, constrained, and solved exactly like any other, and if the model is nonlinear the step is driven by the Newton iteration.
Explicit time integration¶
A few models also offer explicit integration, where the time derivative is evaluated at the known current state, so the new state follows without solving a system involving the stiffness matrix \(A\). Forward Euler, for example, gives
and the mass matrix \(M\) is usually lumped to a diagonal so that \(M^{-1}\) is trivial. Explicit steps are very cheap individually but are stable only below a critical step size (a CFL-type limit), which makes them a poor fit for the stiff, diffusive problems that dominate μfem. Only a small number of models support explicit integration, typically wave-like formulations; implicit backward Euler is the default.
Steps and inner iterations¶
The runner repeats this for each step, using the just-computed \(\mathbf{x}^{n+1}\) as the history for the next one. Within a single step it may take several inner iterations to resolve material nonlinearity or coupling between models before moving on. The choice of \(\Delta t\) trades accuracy against cost: smaller steps follow fast transients more faithfully, larger steps reach the final time sooner.