Hybrid Time Integration Algorithms for Co-Simulation of Multiscale Multiphysics Systems

Computer simulations of very complex systems are central to many fields in science and engineering, including mechanical and chemical engineering, aeronautics, astrophysics, plasma physics, meteorology and oceanography, finance, environmental sciences, and urban modeling. However, these simulations are hampered by the limitations of currently available numerical methodologies. Specifically, complex systems are driven by multiple simultaneous physical processes with different dynamic characteristics, e.g., atmospheric chemistry and atmospheric transport. Consequently, different components evolve at different rates, some very fast (e.g., concentrations of chemical tracers) and some very slow (e.g., ocean temperature). Traditional numerical methods are ill-suited to solve complex systems with multiple scales and multiple dynamics. This project develops new numerical algorithms that solve different complex system components with different discretizations and different time steps. This new approach will allow accurate and efficient simulations of complex systems and will positively impact many fields in science and engineering. A novel hybrid time integration framework will be constructed to co-simulate complex systems governed by time-dependent partial differential equations. The particular innovation of the hybrid methodology is that it combines discrete and continuous internal stages during each integration step. The mathematical framework offers local truncation error estimates (unlike operator splitting), and provides solutions that do not depend on the convergence of an outer iteration process (unlike relaxation). It allows us to build methods with a higher order of accuracy than current co-simulation methodologies. The developed hybrid methods will have higher orders of accuracy than current co-simulation methodologies while offering tremendous implementation flexibility. High-quality implementations of the new methods will be made available to the community at large. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.