S-fraction multiscale finite-volume method for spectrally accurate wave   propagation
  We develop a method for numerical time-domain wave propagation based on the model order reduction approach. The method is built with high-performance computing (HPC) implementation in mind that implies a high level of parallelism and greatly reduced communication requirements compared to the traditional high-order finite-difference time-domain (FDTD) methods. The approach is inherently multiscale, with a reference fine grid model being split into subdomains. For each subdomain the coarse scale reduced order models (ROMs) are precomputed off-line in a parallel manner. The ROMs approximate the Neumann-to-Dirichlet (NtD) maps with high (spectral) accuracy and are used to couple the adjacent subdomains on the shared boundaries. The on-line part of the method is an explicit time stepping with the coupled ROMs. To lower the on-line computation cost the reduced order spatial operator is sparsified by transforming to a matrix Stieltjes continued fraction (S-fraction) form. The on-line communication costs are also reduced due to the ROM NtD map approximation properties. Another source of performance improvement is the time step length. Properly chosen ROMs substantially improve the Courant-Friedrichs-Lewy (CFL) condition. This allows the CFL time step to approach the Nyquist limit, which is typically unattainable with traditional schemes that have the CFL time step much smaller than the Nyquist sampling rate. 
Vladimir Druskin, Alexander V. Mamonov and Mikhail Zaslavsky
math.NA