18 research outputs found
Discrete Lie Advection of Differential Forms
In this paper, we present a numerical technique for performing Lie advection
of arbitrary differential forms. Leveraging advances in high-resolution finite
volume methods for scalar hyperbolic conservation laws, we first discretize the
interior product (also called contraction) through integrals over Eulerian
approximations of extrusions. This, along with Cartan's homotopy formula and a
discrete exterior derivative, can then be used to derive a discrete Lie
derivative. The usefulness of this operator is demonstrated through the
numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC
Structure-Preserving Discretization of Incompressible Fluids
The geometric nature of Euler fluids has been clearly identified and
extensively studied over the years, culminating with Lagrangian and Hamiltonian
descriptions of fluid dynamics where the configuration space is defined as the
volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed
as a consequence of Noether's theorem associated with the particle relabeling
symmetry of fluid mechanics. However computational approaches to fluid
mechanics have been largely derived from a numerical-analytic point of view,
and are rarely designed with structure preservation in mind, and often suffer
from spurious numerical artifacts such as energy and circulation drift. In
contrast, this paper geometrically derives discrete equations of motion for
fluid dynamics from first principles in a purely Eulerian form. Our approach
approximates the group of volume-preserving diffeomorphisms using a finite
dimensional Lie group, and associated discrete Euler equations are derived from
a variational principle with non-holonomic constraints. The resulting discrete
equations of motion yield a structure-preserving time integrator with good
long-term energy behavior and for which an exact discrete Kelvin's circulation
theorem holds
Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus
We propose a new class of finite element approximations to ideal compressible
magnetohydrodynamic equations in smooth regime. Following variational
approximations developed for fluid models in the last decade, our
discretizations are built via a discrete variational principle mimicking the
continuous Euler-Poincar\'e principle, and to further exploit the geometrical
structure of the problem, vector fields are represented by their action as Lie
derivatives on differential forms of any degree. The resulting semi-discrete
approximations are shown to conserve the total mass, entropy and energy of the
solutions for a wide class of finite element approximations. In addition, the
divergence-free nature of the magnetic field is preserved in a pointwise sense
and a time discretization is proposed, preserving those invariants and giving a
reversible scheme at the fully discrete level. Numerical simulations are
conducted to verify the accuracy of our approach and its ability to preserve
the invariants for several test problems.Comment: 35 pages, 8 figure
Deep Learning for Stable Monotone Dynamical Systems
Monotone systems, originating from real-world (e.g., biological or chemical)
applications, are a class of dynamical systems that preserves a partial order
of system states over time. In this work, we introduce a feedforward neural
networks (FNNs)-based method to learn the dynamics of unknown stable nonlinear
monotone systems. We propose the use of nonnegative neural networks and batch
normalization, which in general enables the FNNs to capture the monotonicity
conditions without reducing the expressiveness. To concurrently ensure
stability during training, we adopt an alternating learning method to
simultaneously learn the system dynamics and corresponding Lyapunov function,
while exploiting monotonicity of the system.~The combination of the
monotonicity and stability constraints ensures that the learned dynamics
preserves both properties, while significantly reducing learning errors.
Finally, our techniques are evaluated on two complex biological and chemical
systems
Geometric, Variational Discretization of Continuum Theories
This study derives geometric, variational discretizations of continuum
theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the
dynamics of complex fluids. A central role in these discretizations is played
by the geometric formulation of fluid dynamics, which views solutions to the
governing equations for perfect fluid flow as geodesics on the group of
volume-preserving diffeomorphisms of the fluid domain. Inspired by this
framework, we construct a finite-dimensional approximation to the
diffeomorphism group and its Lie algebra, thereby permitting a variational
temporal discretization of geodesics on the spatially discretized
diffeomorphism group. The extension to MHD and complex fluid flow is then made
through an appeal to the theory of Euler-Poincar\'{e} systems with advection,
which provides a generalization of the variational formulation of ideal fluid
flow to fluids with one or more advected parameters. Upon deriving a family of
structured integrators for these systems, we test their performance via a
numerical implementation of the update schemes on a cartesian grid. Among the
hallmarks of these new numerical methods are exact preservation of momenta
arising from symmetries, automatic satisfaction of solenoidal constraints on
vector fields, good long-term energy behavior, robustness with respect to the
spatial and temporal resolution of the discretization, and applicability to
irregular meshes
Recommended from our members
Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws
The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling.
Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms