801 Recent Advances in Solving Stiff Phenomena in Computation Mechanics

Dominik Michels, KAUST
Mayya Tokman, UC Merced
David Keyes, KAUST
For most problems, simulating dynamics more accurately invariably means sacrificing computational efficiency. This is especially the case, when the underlying differential equations are stiff. Such systems of equations are characterized by a wide range of time scales present in their evolution. Stiffness arises when the time scale of interest in the dynamics is much slower than the fastest modes of the system. Stiff equations are ubiquitous in a wide range of fields including computational mechanics. Prominent examples include the dynamics of cloth, fibers, fluids, or solids, and their interaction with each other including collision and (frictional) contact handling.

The numerical time integration of stiff systems of differential equations is one of the central problems in numerical analysis. The history of this branch of numerical analysis has been dominated by two classes of time integrators: explicit and implicit. Both types of integrators allow advancing the numerical solution along a discretized time interval, but the numerical properties of these two classes are fundamentally different. Explicit methods require the least amount of computations per time step but suffer severe stability restrictions that limit the allowable size of the time step. Implicit methods possess better stability properties and allow for accurate integration with a much larger time step, but the increase in time step size comes at the expense of significantly more computations required in each time iteration. As the stiffness of the problem grows, integrating equations explicitly over a long period of time becomes impractical and modelers turn to implicit methods. However, implicit schemes are not immune to the increase in stiffness and the amount of computation required per time step grows correspondingly.

Recently, exponential methods emerged as a viable alternative to implicit schemes for a number of stiff problems. A range of exponential integrators have been developed including stiffly accurate methods that are particularly suited for the integration of stiff systems. Moreover, significant progress has been achieved in the development of modern Krylov subspace projection methods.

In this workshop, we intend to discuss recent developments in solving stiff differential equations as state-of-the-art exponential integrators and modern Krylov subspace projection methods. This is tailored to the particular needs of applications in computational mechanics.

Keywords: deformable body mechanics, exponential integrators, Krylov subspace projection methods, stiff phenomena.