##  [Multiple-Rate Integrators for Differential Equations](/archives/ldrd-annual-2019/high-performance-computing/17-ERD-035) 

##   *Carol Woodward*  |  17-ERD-035  

###  Overview 

 This project investigated multirate time-integration methods, which evolve a differential equation system in time by advancing parts of a problem with fast time scales at small time steps and parts of a problem with slow time scales with larger time steps. We developed new methods and theory as well as a set of prototype software implementations. Multirate methods were tested on combustion and climate applications.

###  Impact on Mission 

 This project advances Lawrence Livermore National Laboratory’s core competency in high-performance computing, simulation, and data science. Specifically, it increased Laboratory expertise in time-integration methods and software for multiphysics and multiscale problems found in climate, combustion, astrophysics, and other topics throughout the Department of Energy complex, especially among exascale applications. This project provided the necessary research foundation that, along with later implementations, will move the Laboratory's SUNDIALS library to the forefront of packages for multiphysics applications. In addition, the project allowed the Laboratory's time-integration group to strengthen ties among academic researchers.

###  Publications, Presentations, Etc. 

 Copeland, A., et al. 2017. "Time Integration Methods for the MG2 Microphysics of a Climate Model." Lawrence Livermore National Laboratory Summer Student Poster Symposium, Livermore, CA. LLNL-POST-735408.

 Dallerit, V. and J. Loffeld. 2018. "Towards the Understanding of Multirate Schemes on Adaptive Mesh Refinement Grids." Lawrence Livermore National Laboratory Summer Student Poster Symposium, Livermore, CA. LLNL-POST-755557.

 Dallerit, V., et al. 2019. "New Multirate Implicit-Exponential (IMEXP) Methods for Partitioned Nonlinear Differential Equations." Lawrence Livermore National Laboratory Summer Student Poster Symposium, Livermore, CA. LLNL-POST-782248.

 Loffeld, J., et al. 2018. "Challenges of Constructing Stable Implicit Multirate Methods." Integrating the Integrators Workshop, Banff, Canada. LLNL-PRES-763459.

  –––  . 2018. "Towards a Multirate Numerical Integration Software Suite for General Multiphysics Problems." SIAM Conference on Parallel Processing, Tokyo, Japan. LLNL-PRES-932514.

  –––  . 2019. "Experiences Applying Multirate Methods to a 1D Unsteady Flame Code." International Conference on Scientific Computing and Differential Equations, Innsbruck, Austria. LLNL-PRES-782164.

  –––  . 2019. "Implicit Multirate GARK Methods for Stiff, Multiscale ODEs." Lawrence Livermore National Laboratory Summer Student Poster Symposium, Livermore, CA. LLNL-POST-782247.

  –––  . 2019. "Implicit Multirate GARK Methods." International Conference on Scientific Computing and Differential Equations, Innsbruck, Austria. LLNL-PRES-795217.

 Stewart, J., et al. 2019. "Multi-Rate Methods for Chemical Simulations." Lawrence Livermore National Laboratory Summer Student Poster Symposium, Livermore, CA. LLNL-POST-782252.

 Woodward, C. 2018. "Coupled Systems, Numerical Libraries, and High Performance Computing: How Do We Bring These Together?" International Conference on Terrestrial Systems Research: Monitoring, Prediction and High Performance Computing, Bonn, Germany. LLNL-PRES-748984.

 Woodward, C., et al. 2017. "Looking Toward the Exascale with SUNDIALS Time Integrators." Lawrence Livermore National Laboratory Computation Directorate External Review Committee. LLNL-POST-728895.

  –––  . 2018. "An Overview of the SUNDIALS Suite of Time Integrators and Nonlinear Solvers with a Look Toward Exascale Computing," SIAM Annual Meeting, Portland, OR. LLNL-PRES-754467.

  –––  . 2019. "Multirate Time Integration for Cloud Microphysics." Lawrence Livermore National Laboratory Summer Student Poster Symposium, Livermore, CA. LLNL-POST-782119.