Nonintrusive and Structure Preserving Multiscale Integration of Stiff ODEs, SDEs, and Hamiltonian Systems with Hidden Slow Dynamics via Flow Averaging

Abstract

We introduce a new class of integrators for stiff ODEs as well as SDEs. Examples of subclasses of systems that we treat are ODEs and SDEs that are sums of two terms, one of which has large coefficients. These integrators are (i) Multiscale: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) Versatile: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for identifying explicitly (or numerically) the slow or fast variables (iii) Nonintrusive: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale (iv) Convergent over two scales: strongly over slow processes and in the sense of measures over fast ones. We introduce the related notion of two-scale flow convergence and analyze the convergence of these integrators under the induced topology (v) Structure preserving: They inherit the structure preserving properties of the legacy integrators from which they are derived. Therefore, for stiff Hamiltonian systems (possibly on manifolds), they can be made to be symplectic, time-reversible, and symmetry preserving (symmetries are group actions that leave the system invariant) in all variables. They are explicit and applicable to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over four orders of magnitude of time scales. For stiff Langevin equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction.

A semi-analytical approach to molecular dynamics

“…For instance, the use of multiple time steps in the Verlet-1/r-RESPA/Impulse MTS method can be incorporated as well to better deal with the non-linear part of the force field-although the savings will be unlikely to affect the efficiency of our approach in a significant way since we already addressed the stiffest part. Similarly, Langevin dynamics [32] could be used to replace solvent forces by homogenized and stochastic forces while preserving symplecticity [56]. Finally, we also plan to test out an approach to obtain coarse-grained parallelism using rational Krylov subspaces instead: the associated rational Arnoldi method, in which a linear system is solved independently for each pole of the polynomial denominator [20], could offer a coarse-grained way to parallelize the matrix function computations involved in the integrator.…”

Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs

“…We also refer to [18,25] for multi-scale transport equations and hyperbolic systems of conservation laws with stiff diffusive relaxation. Well-identified slow variables can be simulated with large time-steps using the two-scale structure of the original stiff PDEs (we refer to [1] and [12] for existing examples; slow variables satisfy a non-stiff PDE that can be identified in analogy to equations (A.9) and (A.13) of [32]; we also refer to [14] for a definition of slow variables).…”

“…In this paper, we consider the second category of PDEs and propose a generalization of FLow AVeraging integratORS (FLAVORS) (introduced in [32] for stiff ODEs and SDEs) to stiff PDEs. Multi-scale integrators for stiff PDEs are obtained without the identification of slow variables by turning on and off stiff coefficients in single-step (legacy) integrators (used as black boxes) and alternating microscopic and mesoscopic time steps (Subsection 2.2).…”

“…We illustrate the generality of the proposed strategy by applying it to finite difference methods in Section 2, multi-symplectic integrators in Section 3, and pseudospectral methods in Section 4 (although we have not done so in this paper, the proposed strategy can also be applied to finite element methods or finite volume methods). The convergence of the proposed strategy, after semi-discretization in space, is analyzed in Subsection 5.1, where a non-asymptotic error bound indicates the two-scale convergence ( [32], i.e., strong with respect to hidden slow variables and weak with respect to hidden fast variables) of PDE-FLAVORS. As illustrated by numerical ( Figure 2) and theoretical results (Section 5), an explicit tuning ((h/)…”

“…We present a new class of integrators for stiff PDEs. These integrators are generalizations of FLow AVeraging integratORS (FLAVORS) for stiff ODEs and SDEs introduced in [32] with the following properties: (i) Multiscale: they are based on flow averaging and have a computational cost determined by mesoscopic steps in space and time instead of microscopic steps in space and time; (ii) Versatile: the method is based on averaging the flows of the given PDEs (which may have hidden slow and fast processes). This bypasses the need for identifying explicitly (or numerically) the slow variables or reduced effective PDEs; (iii) Nonintrusive: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale; (iv) Convergent over two scales: strongly over slow processes and in the sense of measures over fast ones; (v) Structure-preserving: for stiff Hamiltonian PDEs (possibly on manifolds), they can be made to be multi-symplectic, symmetry-preserving (symmetries are group actions that leave the system invariant) in all variables and variational.…”