Solving the advection-diffusion equations in biological contexts using the cellular Potts model
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The cellular Potts model (CPM) is a robust, cell-level methodology for simulation of biological tissues and morphogenesis. Both tissue physiology and morphogenesis depend on diffusion of chemical morphogens in the extra-cellular fluid or matrix (ECM). Standard diffusion solvers applied to the cellular potts model use finite difference methods on the underlying CPM lattice. However, these methods produce a diffusing field tied to the underlying lattice, which is inaccurate in many biological situations in which cell or ECM movement causes advection rapid compared to diffusion. Finite difference schemes suffer numerical instabilities solving the resulting advection-diffusion equations. To circumvent these problems we simulate advection diffusion within the framework of the CPM using off-lattice finite-difference methods. We define a set of generalized fluid particles which detach advection and diffusion from the lattice. Diffusion occurs between neighboring fluid particles by local averaging rules which approximate the Laplacian. Directed spin flips in the CPM handle the advective movement of the fluid particles. A constraint on relative velocities in the fluid explicitly accounts for fluid viscosity. We use the CPM to solve various diffusion examples including multiple instantaneous sources, continuous sources, moving sources, and different boundary geometries and conditions to validate our approximation against analytical and established numerical solutions. We also verify the CPM results for Poiseuille flow and Taylor-Aris dispersion.Keywords:
Hagen–Poiseuille equation
Finite difference
Anomalous Diffusion
Lattice (music)
This chapter contains sections titled: Finite Difference Approximations Conditions for Satisfactory Solutions Explicit Finite Difference Method Implicit Finite Difference Methods A Worked Example Comparison of Methods
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A finite-difference seismic modeling method based on triangular grids is presented. Compared with the conventional finite-difference schemes based on rectangular grids, curved velocity boundaries can be more accurately represented by using the new approach. Models with an irregular earth surface can be directly used for seismic modeling without special processing. As the space steps can be varied with velocities and the explicit finite-difference scheme is used for calculation, the method is an effective improvement over the conventional finite-difference methods. Numerical tests demonstrate that this new method is stable, efficient and sufficiently accurate.
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PreviousNext No AccessIntroduction to Petroleum Seismology, Second editionAppendix G: The Explicit and Implicit Finite-Difference Methodshttps://doi.org/10.1190/1.9781560803447.appg SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Abstract As described in Chapter 17, the finite-difference method allows us to approximately solve partial differential equations and integral equations. There are basically two types of finite-difference method: the explicit finite-difference method and the implicit finite-difference method. Chapter 17 describes the explicit finite-difference method. This appendix describes the implicit finite-difference method for the heat equation and contrasts it to the explicit finite-difference method. The implicit finite-difference method of the heat equation is considered because of its importance in microseismicity studies. Table of Contents The heat (or diffusion) equation The explicit finite-difference method The implicit finite-difference method Permalink: https://doi.org/10.1190/1.9781560803447.appgFiguresReferencesRelatedDetails Introduction to Petroleum Seismology, Second editionISBN (print):978-1-56080-343-0ISBN (online):978-1-56080-344-7Copyright: 2018 Pages: 1402 publication data© 2018 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transcribed in any form or by any means, electronic or mechanical, including photocopying and recording, without prior written permission of the publisher.Publisher:Society of Exploration Geophysicists HistoryPublished in print: 01 Jan 2018 CITATION INFORMATION (2018), "Appendix G: The Explicit and Implicit Finite-Difference Methods," Investigations in Geophysics : 1315-1318. https://doi.org/10.1190/1.9781560803447.appg Plain-Language Summary PDF DownloadLoading ...
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Using finite difference methods to simulate the Black-Scholes pricing model is based on non-dividend paying American put option,and get its numerical solution.Through numerical example,a comparison between implicit finite difference methods and explicit finite difference method is presented.By refinement grid the more exact solution is got,and given some useful conclusion in options trading.
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This research is conducted to model flow of traffic on a one lane roadway by partial differential equation (PDE). Then, Finite Difference Method (FDM) is used to solve one-dimensional traffic flow equation. The Finite Difference Method involved is forward difference and central difference. In this problem, the density of cars with fixed ends is considered. The finite difference method (FDM) proceeds by replacing the derivatives in the traffic flow equations by finite difference approximations. This gives a large algebraic system of equations to be solved, which can be solved easily in mathematics software. MATLAB Distributed Computing R2010a software is used to perform the computational experiment while Microsoft Excel is used toillustrated the graphs. In this research, the effect of different step space, h and step time, k are investigated. Besides, comparison between finite difference solutions and analytical solutions will determine the accuracy of finite difference method (FDM).
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Summary We present an alternative scheme for calculating finite difference coefficients in seismic wavefield modelling. This novel technique seeks to minimise the difference between the accurate value of spatial derivatives and the value calculated by the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method maximises the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods whilst comparable to Zhang’s optimised finite difference method.
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This chapter contains sections titled: Ordinary Finite Difference Methods Improved Finite Difference Methods Finite Difference Analysis of Moderately Thick Plates Advances in Finite Difference Methods Summary and Conclusions Problems
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