Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeometric Discretization of the EPDiff Equations
96
0
0.0
(
0
)
Authors
Dmitry Pavlov
Ask ChatGPT about the research
No Arabic abstract
The main objective of this paper is to develop a general method of geometric discretization for infinite-dimensional systems and apply this method to the EPDiff equation. The method described below extends one developed by Pavlov et al. for incompressible Euler fluids. Here this method is presented in a general case applicable to all, not only divergence-free, vector fields. Also, a different (pseudospectral) representation of the velocity field is used. We will apply this method to the one-dimensional EPDiff equation and present numerical results.
rate research
Read More
In this paper we consider the numerical approximation of systems of Boussinesq-type to model surface wave propagation. Some theoretical properties of these systems (multi-symplectic and Hamiltonian formulations, well-posedness and existence of solitary-wave solutions) were previously analyzed by the authors in Part I. As a second part of the study, considered here is the construction of geometric schemes for the numerical integration. By using the method of lines, the geometric properties, based on the multi-symplectic and Hamiltonian structures, of different strategies for the spatial and time discretizations are discussed and illustrated.
We propose a new semi-discretization scheme to approximate nonlinear Fokker-Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric. We discretize the underlying state by a finite graph and define a discrete 2-Wasserstein metric. Based on such metric, we introduce a dynamical system, which is a gradient flow of the discrete free energy. We prove that the new scheme maintains dissipativity of the free energy and converges to a discrete Gibbs measure at exponential (dissipation) rate. We exhibit these properties on several numerical examples.
We propose and study numerically the implicit approximation in time of the Navier-Stokes equations by a Galerkin-collocation method in time combined with inf-sup stable finite element methods in space. The conceptual basis of the Galerkin-collocation approach is the establishment of a direct connection between the Galerkin method and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs in terms of less complex algebraic systems of the latter. Regularity of higher order in time of the discrete solution is ensured further. As an additional ingredient, we employ Nitsches method to impose all boundary conditions in weak form with the perspective that evolving domains become feasible in the future. We carefully compare the performance poroperties of the Galerkin-collocation approach with a standard continuous Galerkin-Petrov method using piecewise linear polynomials in time, that is algebraically equivalent to the popular Crank-Nicholson scheme. The condition number of the arising linear systems after Newton linearization as well as the reliable approximation of the drag and lift coefficient for laminar flow around a cylinder (DFG flow benchmark with $Re=100$) are investigated. The superiority of the Galerkin-collocation approach over the linear in time, continuous Galerkin-Petrov method is demonstrated therein.
The paper is devoted to discretization of integral norms of functions from a given collection of finite dimensional subspaces. For natural collections of subspaces of the multivariate trigonometric polynomials we construct sets of points, which are optimally (in the sense of order) good for each subspace of a collection from the point of view of the integral norm discretization. We call such sets universal. Our construction of the universal sets is based on deep results on existence of special nets, known as (t,r,d)-nets.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
