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We introduce an efficient algorithm for computing fractional integrals and derivatives and apply it for solving problems of the calculus of variations of fractional order. The proposed approximations are particularly useful for solving…
A common strategy in the numerical solution of partial differential equations is to define a uniform discretization of a tensor-product multi-dimensional logical domain, which is mapped to a physical domain through a given coordinate…
We show how to use numerical methods within the framework of successive scaling to analyse the microstructure of turbulence, in particular to find inertial range exponents and structure functions. The methods are first calibrated on the…
Classical optimization is a cornerstone of the success of variational quantum algorithms, which often require determining the derivatives of the cost function relative to variational parameters. The computation of the cost function and its…
Hyperbolic networks have shown prominent improvements over their Euclidean counterparts in several areas involving hierarchical datasets in various domains such as computer vision, graph analysis, and natural language processing. However,…
Computational fluid dynamics (CFD) is a specialised branch of fluid mechanics that utilises numerical methods and algorithms to solve and analyze fluid-flow problems. One promising avenue to enhance CFD is the use of quantum computing,…
The goal of this study is to develop an efficient numerical algorithm applicable to a wide range of compressible multicomponent flows. Although many highly efficient algorithms have been proposed for simulating each type of the flows, the…
Efficient simulation of the Navier-Stokes equations for fluid flow is a long standing problem in applied mathematics, for which state-of-the-art methods require large compute resources. In this work, we propose a data-driven approach that…
An understanding of the hydrodynamics of multiphase processes is essential for their design and operation. Multiphase computational fluid dynamics (CFD) simulations enable researchers to gain insight which is inaccessible experimentally.…
We propose a variational scheme for computing Wasserstein gradient flows. The scheme builds upon the Jordan--Kinderlehrer--Otto framework with the Benamou-Brenier's dynamic formulation of the quadratic Wasserstein metric and adds a…
Numerical discretisations of partial differential equations (PDEs) can be written as discrete convolutions, which, themselves, are a key tool in AI libraries and used in convolutional neural networks (CNNs). We therefore propose to…
We present a pedagogical review of some of the methods employed in Eulerian computational fluid dynamics (CFD). Fluid mechanics is governed by the Euler equations, which are conservation laws for mass, momentum, and energy. The standard…
In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional…
We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap…
Computational fluctuating hydrodynamics aims at understanding the impact of thermal fluctuations on fluid motions at small scales through numerical exploration. These fluctuations are modeled as stochastic flux terms and incorporated into…
In this work we study arbitrary-order hybrid discretizations of Friedrichs systems. Friedrichs systems provide a framework that goes beyond the standard classification of partial differential equations into hyperbolic or elliptic, and are…
We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles…
Turbulent flows are chaotic and unsteady, but their statistical distribution converges to a statistical steady state. Engineering quantities of interest typically take the form of time-average statistics such as $ \frac{1}{t} \int_0^t f (…
We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are…
A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. For this, a mathematical model is developed to incorporate homogeneous Dirichlet and Neumann type boundary conditions. The…