Time Integration

Tools for Problem Discretization
Time Integration


Developing time integration software tailored to efficient solution of problems arising in the context of simulation and optimization of multiphysics and multiscale simulations.


As scientific models improve in physical fidelity, their simulations increasingly couple large-scale, stiff, and temporally disparate interacting processes. In addition, more complex optimization problems continue to be addressed by DOE SciDAC scientists. FASTMath efforts, therefore, focus on software delivery and application support for implicit and semi-implicit time integration methods, development of next-generation time integrator software for multi-rate problems, and support for efficient solution of optimization problems involving time-dependent simulators. Methods involved in this work include linear multistep methods, additive and generalized additive Runge-Kutta methods, spectral deferred correction methods, and multigrid reduction in time schemes.

Planned work includes the addition of multirate integrators and multigrid reduction in time capabilities to SUNDIALS, multilevel spectral deferred correction methods to the AMReX code base, and efficiency additions to PETsc that exploit linear solver system structures over repeated solves within an adjoint solution framework for optimization.

Recent accomplishments: 

SciDAC-3 FASTMath time integration efforts focused on software delivery and application support for implicit/explicit multistage (Runge-Kutta) integrators, upgrades of these methods to implement multi- and many-core parallelism, and development of next-generation solvers for multirate problems, through investigation of additive Runge-Kutta algorithms and spectral deferred correction methods. The SUNDIALS suite added the ARKode integrator library, including high-order, adaptive time integration with multistage implicit, explicit, and mixed implicit-explicit (ImEx) methods. Based on this framework, we developed a new class of fourth-order, explicit multirate time integration methods with embedded error estimators to enable temporal adaptivity and developed a novel approach to filtering noise from atomistic data within multiscale simulations. In addition, new vector kernels were added to SUNDIALS in support of OpenMP and pThreads-based applications.