TIFR Centre for Applicable Mathematics

Abstract Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form where and are any consistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.

Abstract This paper deals with an improvement of the Trudinger–Moser inequality associated to the embedding of the standard Sobolev space into Orlicz spaces for Ω a smooth bounded domain in ℝ2. The inequality proved here gives in particular precise informations on a previous result obtained by Lions and can be very useful in the study of lack of compactness of the embedding of into {exp(4πu 2) ∈ L 1(Ω)}. We also provide a general asymptotic analysis for sequences of solutions to elliptic PDE's with critical Sobolev growth which blow up at some point. We obtain in particular a result which is well-known in higher dimensions: the concentration points are located at critical points of the regular part of the Green function of the linear operator involved in the equation.

We deal with a single conservation law in one space dimension whose flux function is discontinuous in the space variable and we introduce a proper framework of entropy solutions. We consider a large class of fluxes, namely, fluxes of the convex-convex type and of the concave-convex (mixed) type. The alternative entropy framework that is proposed here is based on a two step approach. In the first step, infinitely many classes of entropy solutions are defined, each associated with an interface connection. We show that each of these class of entropy solutions form a contractive semigroup in L 1 and is hence unique. Godunov type schemes based on solutions of the Riemann problem are designed and shown to converge to each class of these entropy solutions. The second step is to choose one of these classes of solutions. This choice depends on the Physics of the problem being considered and we concentrate on the model of two-phase flows in a heterogeneous porous medium. We define an optimization problem on the set of admissible interface connections and show the existence of an unique optimal connection and its corresponding optimal entropy solution. The optimal entropy solution is consistent with the expected solutions for two-phase flows in heterogeneous porous media.

We prove that a sharp Moser–Trudinger inequality holds true on a conformal disc if and only if the metric is bounded from above by the Poincaré metric. We also derive necessary and sufficient conditions for the validity of a sharp Moser–Trudinger inequality on a simply connected domain in ℝ 2 .

We present a novel active learning algorithm, termed as iterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained optimization problems. This algorithm is based on deep neural networks and its key feature is the iterative selection of training data through a feedback loop between deep neural networks and any underlying standard optimization algorithm. Numerical examples for optimal control, parameter identification and shape optimization problems for PDEs are provided to demonstrate that ISMO significantly outperforms a standard deep neural network based surrogate optimization algorithm as well as standard optimization algorithms.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributors Adimurthi* G. Mancini** S.L. Yadava*

In this paper we study the existence, non-existence and simplicity of the first eigenvalue of the perturbed Hardy-Sobolev operator under various assumptions on the perturbation q . We study the asymptotic behaviour of the first eigenfunction near the origin when the perturbation q is q = s , 0 < s < 1. We will also establish the best constant in a Hardy-Sobolev inequality proved by Adimurthi et al.

Plant stoichiometry, the relative concentration of elements, is a key regulator of ecosystem functioning and is also being altered by human activities. In this paper we sought to understand the global drivers of plant stoichiometry and compare the relative contribution of climatic vs. anthropogenic effects. We addressed this goal by measuring plant elemental (C, N, P and K) responses to eutrophication and vertebrate herbivore exclusion at eighteen sites on six continents. Across sites, climate and atmospheric N deposition emerged as strong predictors of plot-level tissue nutrients, mediated by biomass and plant chemistry. Within sites, fertilization increased total plant nutrient pools, but results were contingent on soil fertility and the proportion of grass biomass relative to other functional types. Total plant nutrient pools diverged strongly in response to herbivore exclusion when fertilized; responses were largest in ungrazed plots at low rainfall, whereas herbivore grazing dampened the plant community nutrient responses to fertilization. Our study highlights (1) the importance of climate in determining plant nutrient concentrations mediated through effects on plant biomass, (2) that eutrophication affects grassland nutrient pools via both soil and atmospheric pathways and (3) that interactions among soils, herbivores and eutrophication drive plant nutrient responses at small scales, especially at water-limited sites.

Solving the Euler equations of ideal hydrodynamics as accurately and efficiently as possible is a key requirement in many astrophysical simulations. It is therefore important to continuously advance the numerical methods implemented in current astrophysical codes, especially also in light of evolving computer technology, which favours certain computational approaches over others. Here we introduce the new adaptive mesh refinement (AMR) code TENET, which employs a high-order discontinuous Galerkin (DG) scheme for hydrodynamics. The Euler equations in this method are solved in a weak formulation with a polynomial basis by means of explicit Runge-Kutta time integration and Gauss-Legendre quadrature. This approach offers significant advantages over commonly employed second-order finite-volume (FV) solvers. In particular, the higher order capability renders it computationally more efficient, in the sense that the same precision can be obtained at significantly less computational cost. Also, the DG scheme inherently conserves angular momentum in regions where no limiting takes place, and it typically produces much smaller numerical diffusion and advection errors than an FV approach. A further advantage lies in a more natural handling of AMR refinement boundaries, where a fall-back to first order can be avoided. Finally, DG requires no wide stencils at high order, and offers an improved data locality and a focus on local computations, which is favourable for current and upcoming highly parallel supercomputers. We describe the formulation and implementation details of our new code, and demonstrate its performance and accuracy with a set of two-and three-dimensional test problems. The results confirm that DG schemes have a high potential for astrophysical applications.

In this paper, it is proved that positive solutions of non linear equation involving the N–Laplacian in a ball in RN with Dirichlet boundary condition are radial and radially decreasing provided that the nonlinearity is a continuous function ƒ(t) (satisfying suitable growth conditions) which is strictly positive for t>0. The method generalizes that of Lions for the Laplacian in two dimensions. The method of the present paper can also be extended to an analogous mixed boundary value problem in a convex cone.

Abstract. We prove a version of the Trudinger–Moser inequality in the hyperbolic space ℍ N , which gives a sharper version of the Trudinger–Moser inequality on the Euclidean unit ball, as well as a hyperbolic space version of the Onofri inequality, and prove the existence of extremal functions to some related problems.

An $L^2$ version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the $\overline {\partial }$-operator is established. This duality is used to study the solution of the $\overline {\partial }$-equation with prescribed support. Applications are given to $\overline {\partial }$-closed extension of forms, as well as to Bochner-Hartogs type extension of CR functions.

Modern astrophysical simulations aim to accurately model an ever-growing array of physical processes, including the interaction of fluids with magnetic fields, under increasingly stringent performance and scalability requirements driven by present-day trends in computing architectures. Discontinuous Galerkin (DG) methods have recently gained some traction in astrophysics, because of their arbitrarily high order and controllable numerical diffusion, combined with attractive characteristics for high-performance computing. In this paper, we describe and test our implementation of a DG scheme for ideal magnetohydrodynamics (MHD) in the arepo-dg code. Our DG-MHD scheme relies on a modal expansion of the solution on Legendre polynomials inside the cells of an Eulerian octree-based adaptive mesh refinement grid. The divergence-free constraint of the magnetic field is enforced using one out of two distinct cell-centred schemes: either a Powell-type scheme based on non-conservative source terms, or a hyperbolic divergence cleaning method. The Powell scheme relies on a basis of locally divergence-free vector polynomials inside each cell to represent the magnetic field. Limiting prescriptions are implemented to ensure non-oscillatory and positive solutions. We show that the resulting scheme is accurate and robust: it can achieve high-order and low numerical diffusion, as well as accurately capture strong MHD shocks. In addition, we show that our scheme exhibits a number of attractive properties for astrophysical simulations, such as lower advection errors and better Galilean invariance at reduced resolution, together with more accurate capturing of barely resolved flow features. We discuss the prospects of our implementation, and DG methods in general, for scalable astrophysical simulations.

Numerical solutions are obtained for the hydro-magnetic mixed convection boundary layer flow of an electrically conducting fluid over a non-isothermal wedge in the presence of variable thermal conductivity. The effects due to viscous dissipation, internal heat generation/absorption, thermal radiation, Joule heating and stress work are included. The governing partial differential equations of the problem, subjected to the appropriate boundary conditions are solved numerically by an efficient finite difference scheme. Numerical calculations are carried out for several sets of values of the dimensionless parameters and a careful study of the results obtained reveal that the flow field is influenced appreciably by the applied magnetic field in addition to the other parameters. Numerical results for the velocity and temperature fields, the local skin-friction coefficient and the local Nusselt number are presented graphically and discussed. To validate the numerical method, comparisons are made with the available results in the literature as special cases and the results are found to be in good agreement. The results obtained reveal many interesting behaviors that warrant further study of the flow and heat transfer characteristics over the permeable wedge.

The momentum ray transform $ I^k $ integrates a rank $ m $ symmetric tensor field $ f $ over lines in $ \mathbb{R}^n $ with the weight $ t^k $: $ (I^k\!f)(x,\xi) = \int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\, \mathrm{d} t. $ In particular, the ray transform $ I = I^0 $ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $ f $ from the data $ (I^0\!f,I^1\!f,\dots, I^m\!f) $. In the cases of $ m = 1 $ and $ m = 2 $, we derive the Reshetnyak formula that expresses $ \|f\|_{H^s_t({\mathbb R}^n)} $ through some norm of $ (I^0\!f,I^1\!f,\dots, I^m\!f) $. The $ H^{s}_{t} $-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

Micro fabricated fluidic devices provide an accessible micro-environment for in vivo studies on small organisms. Simple fabrication processes are available for microfluidic devices using soft lithography techniques. Microfluidic devices have been used for sub-cellular imaging, in vivo laser microsurgery and cellular imaging. In vivo imaging requires immobilization of organisms. This has been achieved using suction, tapered channels, deformable membranes, suction with additional cooling anesthetic gas, temperature sensitive gels, cyanoacrylate glue and anesthetics such as levamisole. Commonly used anesthetics influence synaptic transmission and are known to have detrimental effects on sub-cellular neuronal transport. In this study we demonstrate a membrane based poly-dimethyl-siloxane (PDMS) device that allows anesthetic free immobilization of intact genetic model organisms such as Caenorhabditis elegans (C. elegans), Drosophila larvae and zebrafish larvae. These model organisms are suitable for in vivo studies in microfluidic devices because of their small diameters and optically transparent or translucent bodies. Body diameters range from -10 μm to -800 μm for early larval stages of C. elegans and zebrafish larvae and require microfluidic devices of different sizes to achieve complete immobilization for high resolution time-lapse imaging. These organisms are immobilized using pressure applied by compressed nitrogen gas through a liquid column and imaged using an inverted microscope. Animals released from the trap return to normal locomotion within 10 min. We demonstrate four applications of time-lapse imaging in C. elegans namely, imaging mitochondrial transport in neurons, pre-synaptic vesicle transport in a transport-defective mutant, glutamate receptor transport and Q neuroblast cell division. Data obtained from such movies show that microfluidic immobilization is a useful and accurate means of acquiring in vivo data of cellular and sub-cellular events when compared to anesthetized animals (Figure 1J and 3C-F). Device dimensions were altered to allow time-lapse imaging of different stages of C. elegans, first instar Drosophila larvae and zebrafish larvae. Transport of vesicles marked with synaptotagmin tagged with GFP (syt.eGFP) in sensory neurons shows directed motion of synaptic vesicle markers expressed in cholinergic sensory neurons in intact first instar Drosophila larvae. A similar device has been used to carry out time-lapse imaging of heartbeat in -30 hr post fertilization (hpf) zebrafish larvae. These data show that the simple devices we have developed can be applied to a variety of model systems to study several cell biological and developmental phenomena in vivo.

Parameter identification is studied for infinite dimensional linear systems. An almost sure characterization of sample path-wise limit sets of maximum likelihood estimates is given.

We consider the scalar conservation law with strict convex flux in one space dimension. In this paperwe study the exact controllability of entropy solution by using initial or boundary data control. Some partial results have been obtained in [5],[23]. Here we investigate the precise conditions under which, the exact controllability problem admits a solution.The basic ingredients in the proof of these results are, Lax-Oleinik [15] explicit formula and finer properties of thecharacteristics curves.

The aim of this work is to demonstrate a curious property of general periodic structures. It is well known that the corresponding homogenized matrix is positive definite. We calculate here the next order Burnett coefficients associated with such structures. We prove that these coefficients form a tensor which is negative semidefinite. We also provide some examples showing degeneracy in multidimension.

In this paper, we study the Hardy–Rellich inequalities for polyharmonic operators in the critical dimension and an analogue in the p-biharmonic case. We also develop some optimal weighted Hardy–Sobolev inequalities in the general case and discuss the related eigenvalue problem. We also prove W 2,q (Ω) estimates in the biharmonic case.