Finite Volume Methods: Foundation and Analysis

Abstract

Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

Modified shallow water equations for significantly varying seabeds

“…In this study, we choose a simple shallow water ansatz, which is a velocity field and velocity potential independent of the vertical coordinate y such that 6) whereū(x, t) is the depth-averaged horizontal velocity andv(x, t) is the vertical velocity at the bottom. In this ansatz, we take for simplicity the pseudo-velocities to be equal to the velocity field u = µ, v = ν.…”

“…are reconstructions of conservative variablesw from left and right sides of each cell interface [6,60]. The reconstruction procedure employed in the present study is described below.…”

“…In other words, the solution is reconstructed on the cells while the solution gradient is estimated on the dual mesh as it is often performed in more modern schemes [6]. A brief summary of the UNO2 reconstruction can be also found in [24].…”

A-posteriori error analysis for lithium-ion concentrations in batteries utilizing the reduced-basis method

“…We discretize (1) by the classical FV method; see, e.g., [3]. The integration over time is realized by the backward Euler method.…”

“…The discretization of the nonlinear µPDE using the (cell-centered) finite volume (FV) techniques (see, e.g., [3]), leads to very large systems that are expensive to solve. The goal is to develop a reduced-order model for the parametrized PDE that is cheap to evaluate.…”