Classical methods solve in each time step for all unknowns at once. This may become very inefficient or impossible for large systems of equations, in particular if the system stems from a coupled problem. This is typically a system of (partial-differential-algebraic) equations that consists of subproblems with different properties. Often the subproblems describe different physical effects (multiphysics), for example electromagnetic and heat distribution. The subproblems are mutually connected by coupling conditions (connecting “inputs” and “outputs”). Often, the various phenomenae evolve at different time and spatial scales (multiscale).

Let us consider a highly integrated electric circuit. It produces heat, which in turn affects its behavior as an electrical system. One must couple electric and thermal subproblem descriptions. On the one hand, this creates multiple time scales due to different speeds of distribution (multirate). On the other hand, in a professional environment one usually has dedicated tools for each subproblem and the overall problem formulation is not available.

Hence, separate solvers that exchange data according to the coupling conditions tackle the subproblems individually. This is called “co-operative simulation” (co-simulation). The simulation of the subproblems is organized either in parallel or sequentially. The solutions are only exchanged at synchronization points and thus the coupling is often called “weak”. If the exchange is done repeatedly on time windows, this approach yields a dynamic iteration scheme (sometimes called waveform relaxation).

Dynamic iteration is a generalization of the static iteration methods for linear equations (e.g. the classical “Gauß-Seidel” method). Typically the time interval of interest is split into smaller windows (macro steps), on which the individual subproblems are solved by a time-stepping method (using micro steps). This allows us to apply a different, i.e. “multirate” step sizes in the process of solving each subproblem. After exchanging information, the simulation of each subproblem is repeated until convergence is reached (sweep).

Convergence is guaranteed for problems consisting of ordinary differential equations, but not for all differential-algebraic equations. Current research involves a deeper understanding of the order of convergence, the development of an efficient window/sweep control and the usage of model order reduction techniques within the framework of dynamic iteration.

Exemplary field/circuit simulation

Simulation of a field/circuit coupled problem. The window size is choosen as 0.01s and 5 iterations (sweeps) are performed per window.

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