Arbitrarily high-order (weighted) essentially non-oscillatory finite difference schemes for anelastic flows on staggered meshes

We propose a WENO finite difference scheme to approximate anelastic flows, and scalars advected by them, on staggered grids. In contrast to existing WENO schemes on staggered grids, the proposed scheme is designed to be arbitrarily high-order accurate as it judiciously combines ENO interpolations of velocities with WENO reconstructions of spatial derivatives. A set of numerical experiments are presented to demonstrate the increase in accuracy and robustness with the proposed scheme, when compared to existing WENO schemes and state-of-the-art central finite difference schemes

Statistical Solutions for the Incompressible Euler Equations with Finite Volume Methods

The theoretical understanding of the fundamental equations of fluid dynamics, the Navier-Stokes and Euler equations, is currently limited, and results of existence and uniqueness of their solutions are not known in a general setting. Statistical solutions have been recently proposed as a suitable framework for the formal treatment of turbulence. These are defined as a probability measure in a space of integrable functions, which verify an infinite hierarchy of partial differential equations in a statistical sense. Statistical solutions can be efficiently approximated by Monte Carlo numerical methods. The main result of this work is the proof that, under experimentally verifiable conditions, a finite volume scheme based on a discrete Leray projection produces sequences which converge, in a weak topology, to statistical solutions of the incompressible Euler equations. We support this theoretical work with a range of numerical examples that showcase the validity of these assumptions, convergence, robustness of the approach with respect to the underlying numerical method, and other relevant statistical properties of the flow. We additionally present experimental results obtained with a novel numerical scheme for the approximation of statistical solutions, based on spatially high-order finite difference schemes, both central and (W)ENO

Statistical solutions of the incompressible Euler equations

We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose time-evolution is determined from the underlying Euler equations. We prove partial well-posedness results for dissipative statistical solutions and propose a Monte Carlo type algorithm, based on spectral viscosity spatial discretizations, to approximate them. Under verifiable hypotheses on the computations, we prove that the approximations converge to a statistical solution in a suitable topology. In particular, multi-point statistical quantities of interest converge on increasing resolution. We present several numerical experiments to illustrate the theory.ISSN:0218-2025ISSN:1793-631

On the conservation of energy in two-dimensional incompressible flows

We prove the conservation of energy for weak and statistical solutions of the two-dimensional Euler equations, generated as strong (in an appropriate topology) limits of the underlying Navier-Stokes equations and a Monte Carlo-Spectral Viscosity numerical approximation, respectively. We characterize this conservation of energy in terms of a uniform decay of the so-called structure function, allowing us to extend existing results on energy conservation. Moreover, we present numerical experiments with a wide variety of initial data to validate our theory and to observe energy conservation in a large class of two-dimensional incompressible flows

Arbitrarily High-Order (Weighted) Essentially Non-Oscillatory Finite Difference Schemes for Anelastic Flows on Staggered Meshes

We propose a WENO finite difference scheme to approximate anelastic flows, and scalars advected by them, on staggered grids. In contrast to existing WENO schemes on staggered grids, the proposed scheme is designed to be arbitrarily high-order accurate as it judiciously combines ENO interpolations of velocities with WENO reconstructions of spatial derivatives. A set of numerical experiments are presented to demonstrate the increase in accuracy and robustness with the proposed scheme, when compared to existing WENO schemes and state-of-the-art central finite difference schemes. © 2021 Global-Science PressISSN:1991-7120ISSN:1815-240