Mathematics Colloquia and Seminars

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Conventional explicit integrators for ODEs are notoriously inefficient for numerically integrating stiff initial-value problems. We introduce the computational technique of "virtually" preconditioning a stiff system so that an outer explicit integrator can be used to solve an equivalent, nonstiff system. The preconditioning is accomplished by using an inner layer of explicit integrators that efficiently develop and stabilize the system at fast and intermediate time scales. The outer integrator can then take accurate time steps commensurate with the slow time scales for the multiscale problem.

This technical approach has significant benefits for high-performance computing because the integrators are explicit methods that scale well on massively parallel machines. It is also a cornerstone for enabling the Equation-Free and Heterogeneous Multiscale Method frameworks for solving atomistic-macroscopic problems. We introduce the main ideas behind the approach and outline some of our preliminary results in this direction.