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.

Global existence for a go-or-grow multiscale model for tumor invasion with therapy

“…We use a monotone, E-flux scheme, such as the Godunov method (see e.g., [1,25]), which is given by the following:…”

“…Space discretization is done via the Finite Volume Method (see e.g., [7,1]) and the time discretization is implemented via an explicit one-step Euler method. In our simulations we use for the terms and coefficients in (2.1a)-(2.1e) the following definitions:…”

Topology-Preserved Diffusion Distance for Histogram Comparison

“…The approach is briefly introduced as follows. First, the spatial derivative ∇ · ∇u(x,t) is discretized by finite volume method [1].…”

“…Second, we utilize numerical methods for PDE, i.e. finite volume method [1] and backward Euler scheme [10], to solve the diffusion process. In contrast, diffusion distance uses convolution to approximate the diffusion, which cannot handle the non-trivial topology.…”

Mathematical modeling of the axisymmetric internal ballistics processes

“…Расчет течения газопороховой реагирующей смеси основывается на интегральном представлении определяющей системы уравнений (3) и методе конечных объемов [18]. Предполагая, что Ω -некоторая двумерная замкнутая область в плоскости ( ) , r z , ограниченная границей Γ , уравнения (3) сводятся к следующим интегральным соотношениям:…”