HFSS15: Local Time Stepping

A prime benefit of satisfying Maxwell’s equations in a way analogous to the Finite Volume Method is greatly reduced RAM usage. Since the DGTD solver avoids solving the large matrix equation that represents the entire computational domain at each time step, no large matrix needs to be stored.

One of the key features of the DGTD solver is an innovative local time-stepping scheme that advances in time each mesh element independently from its neighbors. This feature efficiently exploits the use of highly unstructured meshes and hp-adaptivity: in the DGTD solver, each mesh element has its own energy-based stability criterion [12], calculated from its size, material properties and basis-function type and order.

Figure 1 illustrates the process. This figure shows a cut-plane through an unstructured mesh model of a ground-penetrating radar. Each element of the mesh is colored according to the time-step values used to locally advance the fields in time. Large elements are colored yellow and use larger time-steps than the smaller elements colored green and or even smaller elements in blue. This results in fewer iterations for most mesh elements and reduced overall simulation times.

Fig. 1 Local time stepping

The time step of any mesh element is chosen such that at the largest time steps the entire simulation is “in sync.” From a user point of view however, the local time-stepping scheme is transparent and all time-domain data are provided or rendered synchronously “on the fly” as needed. In contrast to FIT or FDTD, the DGTD solver doesn’t advance in time uniformly using the smallest time step but uses an optimal range of time steps that minimize simulation times, ensures stability and minimizes the dispersion error.

One last important feature of DGTD solver is its affinity for parallel computations. This is a direct consequence of the local nature of its formulation that yields a spatially compact numerical scheme. Parallelism is naturally exploited with this scheme by partitioning the computational domain into sub-domains and solving each sub-domain concurrently on multi-core or many-core systems.