Higher-order adaptive methods for fluid dynamics

Abstract : Fluids (gases and liquids) exist everywhere around us. Water covers 70% of the Earth’s crust and gases like nitrogen and oxygen surround the planet. The field of fluid dynamics involves the study of liquids or gases in motion. The equations which govern the motion of fluids viz. the Navier–Stokes equations, are complex non-linear partial differential equations which do not have closed-form analytical solutions for most problems of practical interest. However, using numerical schemes, these partial differential equations of continuous variables can be transformed into huge algebraic systems of discrete variables and solved using high-performance computers. A numerical method solved on a computing device will introduce errors in the final solution, will require a given amount of computational resource like memory and processor, and will take a finite amount of time to reach a solution. Thus the development of more accurate and faster algorithms to numerically model the equations of fluid dynamics is a constantly evolving research field. The present document is dedicated to both the study of existing lower-order numerical algorithms as well as either the implementation of existing or development and implementation of new higher-order algorithms, relevant for solving the incompressible Navier–Stokes equations. The entire work has been carried out on the adaptive Cartesian solver for fluid equations Basilisk. We specifically research solvers for convection–diffusion, Poisson–Helmholtz equations, time-marching schemes, and for the shallow-water equations. We look at adaptive mesh methods for solving these equations and taking the Basilisk implementation of the adaptive wavelet algorithm on a quad-octrees as our starting point, we build a novel higher-order adaptive scheme. A recurring theme throughout this thesis is the comparison in accuracy and computing performance of different higher-order schemes when compared to their lower-order counterparts.
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Rajarshi Roy Chowdhury. Higher-order adaptive methods for fluid dynamics. Mechanics of the fluids [physics.class-ph]. Sorbonne Université, 2018. English. ⟨tel-02056238⟩

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