## Project Title

Simulation of Dynamics in Multiscale and Evolution Models

## PI

Mahadevan Ganesh

## Email

mganesh@mines.edu

## Department

Applied Mathematics & Statistics

## Project summary

The purpose of this research is to develop parallel adaptive spatial multiscale and parallel-in-time HPC algorithms for simulation of critical dynamics in multiscale and evolution models in highly heterogeneous media. When the media well-behaved (with few scales and smooth heterogeneity), the standard Finite Element Method (FEM) produces excellent results (thanks adaptive algorithm developments over the last few decades). However, in cases with highly heterogeneous media (for instance, highly oscillatory, has high contrast, or both) the standard FEM is prohibitively expensive to capture the multiscale phenomena present in the problem, even with adaptive mesh refinement techniques.

Such high-contrast problems arise, for instance, when modeling sub-surface fluid flow. As such, numerical methods that can adequately resolve the multiscale features of the solution with computational effort dominated only by solving several local problems in parallel (and the global problem solved only on a coarse-grid) are very desirable in the petroleum industry and elsewhere. This research is concerned with developing such a new class of adaptive Multiscale Finite Element Method (MsFEM).

The standard time discretization procedure is inherently serial, require complete information in the previous time-step before proceeding to the next. Such standard algorithms are not HPC BlueM type architecture friendly w.r.t. to the time variable and also lead to computational bottleneck for long-time simulation to obtain certain desirable properties (such as equilibrium state) in three dimensional space-time evolution models. We develop an efficient HPC friendly

parallel-in-time computational algorithms to avoid this bottleneck for next generation architecture.

## Commercial software

No commercial codes or packages are used

## Open Source software

All versions of the codes use only standard Open Source libraries, such as MPI, and parallel system solving libraries such as ScaLAPACK, PETSC.