Multiscale Computations of Time Dependent Highly Oscillatory Systems

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Many important systems in science, engineering, and industry are modeled by dynamical systems with solutions that are highly oscillatory relative to the overall time scale. Molecular dynamics computations for materials science and biology are of this form and so is simulation of neural processes. Such systems pose severe challenges to numerical simulations. In direct simulations each oscillation must be resolved over the full time interval, which is highly computationally costly. This research project aims to develop new computational methods that are well suited to simulation of such systems. An example of the infinite-dimensional systems that will be considered is high frequency wave propagation. One goal is improving the quality of seismic imaging. Earlier contributions have successfully handled cases with strong separation of large and small scales in the solutions by only applying microscale simulations locally. The current work aims at reducing this requirement by adaptively monitoring the solution and exploiting parallel-in-time computations. This research will go one step further than the typical analytical or numerical averaging and homogenization approaches to deal with more challenging multiscale problems where there is no clear scale separation. The research will start to establish a new multiscale framework, which will support adaptive application of microscale models in small parts of the computational domain. The computational efficiency will result partially from parallelization in time. The theoretical goal is to establish a mathematical link between averaging theories to existing parallel-in-time computational frameworks and filtering techniques. The practical goal is to prepare the study of seismic wave propagation where the microscale model describes detailed diffraction and the effective macroscales are represented by geometrical optics type models.