Non-intrusive 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. An example of subclass 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 ow averaging and so do not resolve the fast variables but rather employ step-sizes determined by slow variables (ii) Basis: the method is based on averaging the ow 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) Non intrusive: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and turned into one of the integrators in this paper by simply 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 ow convergence and analyze the convergence of these integrators under the induced topology. (v) Structure preserving: For stiff Hamiltonian systems (possibly on manifolds), they are symplectic, time-reversible, and symmetric (under the group action leaving the Hamiltonian invariant) in all variables. They are explicit and apply to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over 4 orders of magnitude of time scales. For stiff Langevin equations, they are symmetric (under a group action), time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction.

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