Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging (0908.1241v2)

Abstract: We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it 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) {\it 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) {\it 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) {\it 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) {\it Structure preserving}: 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.

• The paper introduces a novel class of non-intrusive integrators using flow averaging to efficiently handle stiff ODEs, SDEs, and Hamiltonian systems by focusing on slow dynamics.
• These integrators are non-intrusive, allowing integration with existing methods while preserving system structure and achieving convergence across different timescales.
• Supported by numerical experiments, these methods improve efficiency by scaling computational cost with slow variables, making them suitable for complex multiscale simulations in various fields.

Overview of Multiscale Integration of Stiff Dynamical Systems via Flow Averaging

The paper introduces a novel class of integrators designed to efficiently handle stiff ordinary and stochastic differential equations (ODEs and SDEs), as well as Hamiltonian systems that exhibit hidden slow dynamics. These integrators are based on the principle of flow averaging, specifically targeting systems that can be expressed as sums of rapidly varying and slowly varying components. The core innovation presented involves developing a computational approach that remains accurate even when not fully resolving the rapid fluctuations traditionally required by standard numerical methodologies.

Key characteristics and capabilities of these integrators include:

1. Multiscale: By leveraging flow averaging, the proposed methods bypass the need to completely resolve the fast variables, thus scaling the computational cost primarily with the slow variables. This results in significant improvements in efficiency, particularly in multiscale systems.
2. Versatility: The integrators do not necessitate a pre-identification of slow and fast variables, a requirement typically present in multiscale frameworks. Instead, they focus on preserving the dynamics through averaging the system flows directly.
3. Nonintrusive: These methods can transform existing time-stepping schemes that resolve microscopic timescales into multiscale integrators that are effective over larger timescales. This is achieved by alternately applying and removing large coefficients over microscopic timespans.
4. Convergence: The integrators converge over two scales: strongly for slow processes and in the sense of measures for fast processes. The concept of two-scale flow convergence is introduced to formalize this multiscale capability.
5. Structure Preserving: They inherit structure-preserving properties, including symplecticity and time-reversibility, from the legacy integrators they are derived from. This makes them particularly suitable for Hamiltonian systems with stiff potentials.

The considerable strength of these integrators lies in their ability to maintain stability and accuracy across a wide range of time scales. Strongly supported by numerical experiments, these methods demonstrate robustness across several canonical examples, including the Fermi-Pasta-Ulam problem. These applications exemplify how the integrators can address computational challenges in both deterministic and stochastic settings.

Implications and Future Directions

The development of these integrators has broad implications for both the theoretical understanding and practical computation of multiscale dynamical systems. The potential applications extend into fields requiring accurate long-term simulation of complex physical phenomena, such as molecular dynamics, climate modeling, and systems biology.

From a practical perspective, these integrators provide a powerful toolset for researchers dealing with systems characterized by large, disparate timescales. The methods' nonintrusive nature allows for seamless integration with existing computational frameworks, fostering wide applicability without necessitating substantial alterations to the underlying numerical infrastructure.

Given the flexibility and capability highlighted within the research, future work could explore extensions to systems governed by partial differential equations and further variants of stochastic processes. In addition, the quest for enhanced computational speed and accuracy can continue, particularly by focusing on refining these integrators' functionality within high-dimensional systems and increasing their resilience to parameter uncertainties.

Overall, this research lays a crucial foundation for advancing computational techniques in multiscale dynamical systems, promising to broaden the horizon of feasible simulations under computational constraints.