A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations

In this work we present a mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations that in the limit of vanishing dissipation exactly preserves mass, kinetic energy, enstrophy and total vorticity on unstructured grids. The essential ingredients to achieve this are: (i) a velocity-vorticity formulation in rotational form, (ii) a sequence of function spaces capable of exactly satisfying the divergence free nature of the velocity field, and (iii) a conserving time integrator. Proofs for the exact discrete conservation properties are presented together with numerical test cases on highly irregular grids

The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction

This paper is a very brief introduction to idempotent mathematics and related topics.Comment: 24 pages, 2 figures. An introductory paper to the volume "Idempotent Mathematics and Mathematical Physics" (G.L. Ltvinov, V.P. Maslov, eds.; AMS Contemporary Mathematics, 2005). More misprints correcte

Resolve subgrid microscale interactions to discretise stochastic partial differential equations

Constructing discrete models of stochastic partial differential equations is very delicate. Stochastic centre manifold theory provides novel support for coarse grained, macroscale, spatial discretisations of nonlinear stochastic partial differential or difference equations such as the example of the stochastically forced Burgers' equation. Dividing the physical domain into finite length overlapping elements empowers the approach to resolve fully coupled dynamical interactions between neighbouring elements. The crucial aspect of this approach is that the underlying theory organises the resolution of the vast multitude of subgrid microscale noise processes interacting via the nonlinear dynamics within and between neighbouring elements. Noise processes with coarse structure across a finite element are the most significant noises for the discrete model. Their influence also diffuses away to weakly correlate the noise in the spatial discretisation. Nonlinear interactions have two further consequences: additive forcing generates multiplicative noise in the discretisation; and effectively new noise processes appear in the macroscale discretisation. The techniques and theory developed here may be applied to soundly discretise many dissipative stochastic partial differential and difference equations.Comment: Revise

The Novikov Conjecture

We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. .Comment: 16 pages. Dedicated to Sergei Novikov on the occasion of his 80th birthday. To appear in Russian Math Survey, 2019. arXiv admin note: text overlap with arXiv:1811.0208

Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids

In this paper two new families of arbitrary high order accurate spectral DG finite element methods are derived on staggered Cartesian grids for the solution of the inc.NS equations in two and three space dimensions. Pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on a spatially staggered mesh. In the first family, h.o. of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived for the pressure gradient in the momentum equation. The resulting linear system for the pressure is symmetric and positive definite and either block 5-diagonal (2D) or block 7-diagonal (3D) and can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The use of a preconditioner was not necessary. This is a rather unique feature among existing implicit DG schemes for the NS equations. In order to avoid a stability restriction due to the viscous terms, the latter are discretized implicitly. The second family of staggered DG schemes achieves h.o. of accuracy also in time by expressing the numerical solution in terms of piecewise space-time polynomials. In order to circumvent the low order of accuracy of the adopted fractional stepping, a simple iterative Picard procedure is introduced. In this manner, the symmetry and positive definiteness of the pressure system are not compromised. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary h.o. accurate in both space and time. The new numerical method has been thoroughly validated for approximation polynomials of degree up to N=11, using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist.Comment: 46 pages, 15 figures, 4 table

Matrix Analysis of Tracer Transport

We review matrix methods as applied to tracer transport. Because tracer transport is linear, matrix methods are an ideal fit for the problem. A gridded, Eulerian tracer simulation can be approximated as a system of linear ordinary differential equations (ODEs). The first-order stretching and deformation of Lagrangian space can also be calculated using a system of linear ODEs. Solutions to these equations are reviewed as well as special properties. Using matrices to model Eulerian tracer transport can also help understand and improve the stability of numerical solutions. Detailed derivations are included.Comment: Revision for submission to Linear Algebra and Application

Complete enumeration of small realizable oriented matroids

Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e. non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configurations of 9 points and 5-dimensional configurations of 9 points. We could also determine all possible combinatorial types of 5-polytopes with 9 vertices.Comment: 19 pages, 2 figure

Resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations

Coarse grained, macroscale, spatial discretisations of nonlinear nonautonomous partial differential\difference equations are given novel support by centre manifold theory. Dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements. The crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements. The techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential\difference equations, such as the forced Burgers' equation, adopted here as an illustrative example.Comment: arXiv admin note: substantial text overlap with arXiv:math/060170

Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

In this paper we present a novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space-time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order \aposteriori sub-cell ADER-WENO finite volume \emph{limiter}. Notoriously, the original DG method produces strong oscillations in the presence of discontinuous solutions and several types of limiters have been introduced over the years to cope with this problem. Following the innovative idea recently proposed in \cite{Dumbser2014}, the discrete solution within the troubled cells is \textit{recomputed} by scattering the DG polynomial at the previous time step onto a suitable number of sub-cells along each direction. Relying on the robustness of classical finite volume WENO schemes, the sub-cell averages are recomputed and then gathered back into the DG polynomials over the main grid. In this paper this approach is implemented for the first time within a space-time adaptive AMR framework in two and three space dimensions, after assuring the proper averaging and projection between sub-cells that belong to different levels of refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. The spectacular resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, both for the Euler equations of compressible gas dynamics and for the magnetohydrodynamics (MHD) equations.Comment: Computers and Fluids 118 (2015) 204-22

A Physical Perspective on Control Points and Polar Forms: B\'ezier Curves, Angular Momentum and Harmonic Oscillators

Bernstein polynomials and B\'ezier curves play an important role in computer-aided geometric design and numerical analysis, and their study relates to mathematical fields such as abstract algebra, algebraic geometry and probability theory. We describe a theoretical framework that incorporates the different aspects of the Bernstein-B\'ezier theory, based on concepts from theoretical physics. We relate B\'ezier curves to the theory of angular momentum in both classical and quantum mechanics, and describe physical analogues of various properties of B\'ezier curves -- such as their connection with polar forms -- in the context of quantum spin systems. This previously unexplored relationship between geometric design and theoretical physics is established using the mathematical theory of Hamiltonian mechanics and geometric quantization. An alternative description of spin systems in terms of harmonic oscillators serves as a physical analogue of P\'olya's urn models for B\'ezier curves. We relate harmonic oscillators to Poisson curves and the analytical blossom as well. We present an overview of the relevant mathematical and physical concepts, and discuss opportunities for further research.Comment: 22 pages, 14 figures. Comments are welcome