Detecting the birth and death of finite-time coherent sets

Finite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of FTCSs are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter-lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi-material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian, showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent sets.     

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.     

Phase Separation in the Advective Cahn–Hilliard Equation

The Cahn–Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn–Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing, then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection–hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection–diffusion equation.

     

Lagrangian submanifolds and the Hamilton–Jacobi equation

Lagrangian submanifolds are becoming a very essential tool to generalize and geometrically understand results and procedures in the area of mathematical physics. The geometric version of the Hamilton–Jacobi equation in terms of Lagrangian submanifolds enables here some novel interesting applications of the Hamilton–Jacobi equation in holonomic, nonholonomic and time-dependent dynamics from a geometrical point of view.     

On Front Motion in a Burgers-Type Equation with Quadratic and Modular Nonlinearity and Nonlinear Amplification

A singularly perturbed initial–boundary value problem for a parabolic equation known in applications as a Burgers-type or reaction–diffusion–advection equation is considered. An asymptotic approximation of solutions with a moving front is constructed in the case of modular and quadratic nonlinearity and nonlinear amplification. The influence exerted by nonlinear amplification on front propagation and blowing- up is determined. The front localization and the blowing-up time are estimated.

     

Dirichlet duality and the nonlinear Dirichlet problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F (Hess u) = 0 on a smoothly bounded domain Ω ? ?ⁿ. In our approach the equation is replaced by a subset F ? Sym²(?ⁿ) of the symmetric n × n matrices with ?F ? { F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F‐convexity” assumption on the boundary ?Ω. We also study the topological structure of F‐convex domains and prove a theorem of Andreotti‐Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F? that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F? is F, and in the analysis the roles of F and F? are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge‐Ampère equation over ?, ?, and ?; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p‐convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc.     

A locally conservative Eulerian‐Lagrangian control‐volume method for transient advection‐diffusion equations

Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection‐diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian‐Lagrangian control‐volume method (ELCVM) for transient advection‐diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well‐regarded Eulerian‐Lagrangian methods, which were previously shown to be very competitive with many well‐perceived methods. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

Implicit contact dynamics and Hamilton-Jacobi theory

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations.     

Chaotic mixing and fractals in a geophysical jet current

We model Lagrangian lateral mixing and transport of passive scalars in meandering oceanic jet currents by two-dimensional advection equations with a kinematic streamfunction with a time-dependent amplitude of a meander imposed. The advection in such a model is known to be chaotic in a wide range of the meander’s characteristics. We study chaotic transport in a stochastic layer and show that it is anomalous. The geometry of mixing is examined and shown to be fractal-like. The scattering characteristics (trapping time of advected particles and the number of their rotations around elliptical points) are found to have a hierarchical fractal structure as functions of initial particle’s positions. A correspondence between the evolution of material lines in the flow and elements of the fractal is established.     

A uniform optimal‐order estimate for an Eulerian‐Lagrangian discontinuous Galerkin method for transient advection–diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for the Eulerian‐Lagrangian discontinuous Galerkin method for unsteady‐state advection–diffusion equations with general inflow and outflow boundary conditions. It is well‐known that these problems admit dynamic fronts with interior and boundary layers. The estimate holds uniformly with respect to the vanishing diffusion coefficient. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A CFL‐free explicit characteristic interior penalty scheme for linear advection‐reaction equations

We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L² norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions

We give a proof of the theorem of removing isolated singularities of pseudo-holomorphic curves with Lagrangian boundary conditions and bounded symplectic area. The proof is a combination of some L^p-type estimates, standard techniques of geometric P.D.E., and some ideas from symplectic geometry and calibration theory.     

The analytic smoothing effect of solutions for the nonlinear spatially homogeneous Landau equation with hard potentials

In this work, we study the Cauchy problem of the nonlinear spatially homogeneous Landau equation with hard potentials in a close-to-equilibrium framework. We prove that the solution to the Cauchy problem with the initial datum in L² enjoys an analytic regularizing effect, and the evolution of the analytic radius is the same as that of heat equations.

     

Space-time domain decomposition for parabolic problems

We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The discretization of this iteration is the block red-black variant of the waveform relaxation method, and our analysis provides new convergence results for this scheme. The asymptotic convergence rate is super-linear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However it is independent of the number of subdomains, provided the size of the overlap remains fixed. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. Numerical calculations illustrate our analysis.     

Realizability of greedy algorithms

We discuss new models of an “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposal to specify the space-time geometry by the use of the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.