Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Finite Difference Methods for Two-Phase Incompressible Flow in Porous Media

Abstract: Two-phase, incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Convection physically dominates diffusion, and the object of this paper is to develop a finite difference procedure that reflects this dominance. The pressure equation, which is elliptic in appearance, is discretized by a standard five-point difference method. The concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms. A convergence analysis is given for the method.

Numerical solution of an evolution equation with a positive-type memory term

Abstract: We study the numerical solution of an initial-boundary value problem for a Volterra type integro-differential equation, in which the integral operator is a convolution product of a positive-definite kernel and an elliptic partial-differential operator. The equation is discretised in space by the Galerkin finite-element method and in time by finite differences in combination with various quadrature rules which preserve the positive character of the memory term. Special attention is paid to the case of a weakly singular kernel. Error estimates are derived and numerical experiments reported.

A benchmark study of numerical schemes for one‐dimensional arterial blood flow modelling

Abstract: Summary Haemodynamical simulations using one-dimensional (1D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. Recent interest in verifying 1D numerical schemes has led to the development of alternative experimental setups and the use of three-dimensional numerical models to acquire data not easily measured in vivo. In most studies to date, only one particular 1D scheme is tested. In this paper, we present a systematic comparison of six commonly used numerical schemes for 1D blood flow modelling: discontinuous Galerkin, locally conservative Galerkin, Galerkin least-squares finite element method, finite volume method, finite difference MacCormack method and a simplified trapezium rule method. Comparisons are made in a series of six benchmark test cases with an increasing degree of complexity. The accuracy of the numerical schemes is assessed by comparison with theoretical results, three-dimensional numerical data in compatible domains with distensible walls or experimental data in a network of silicone tubes. Results show a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries. All the information used in this study, including the input data for all benchmark cases, experimental data where available and numerical solutions for each scheme, is made publicly available online, providing a comprehensive reference data set to support the development of 1D models and numerical schemes. Copyright © 2015 John Wiley & Sons, Ltd.