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Unformatted text preview: CS 111 - Introduction to Computational Science Solving Diffusion Problems Diffusion phenomena are ubiquitous in science and engineering. For example, diffusion describes the spread of particles through random motion from regions of higher concentration to regions of lower concentration. Consider for instance Oxygen molecules diffusing across cell membranes into cells, and carbon dioxide molecules diffusing out, the diffusion of sugar in a cup or the spread of perfume in a room. The basic equation for the diffusion of a species u is given by: ∂u ∂t = D Δ u + S, (1) where D is the diffusion constant, u is the temperature and S is the source term. The same equation is also valid to describe heat conduction through metals as well as other phenomena such as the effect of viscosity in a fluid. The diffusion equation is also used in Computer Vision to ‘denoise’ noisy images. It is possible to solve the heat equation analytically for very special cases, but more often than not, it is necessary to use computer simulations to solve a typical problem in science and engineering. This module describes the numerical approximations of diffusion problems in one and two spatial dimensions. 1 The Diffusion Equation in 1D node 4 node 3 node 2 node 1 ∆ x node 5 node 6 Figure 1: Discretization of a one dimensional domain with m = 6 nodes. The solution u at the red nodes will be given the value of the boundary conditions through the function BC , while at the blue nodes the solution will be approximated using (3). Consider the heat equation in one spatial dimensions: ∂u ∂t = D ∂ 2 u ∂x 2 + S, (2) 1 CS 111 - Introduction to Computational Science where D is the diffusion constant, u is the temperature and S is the source term. We assume that the value of the temperature is given on the walls of the domain by a function called BC . The source term is also given by a function S . In order to find a numerical solution, we discretize the computational....
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- Fall '08
- Numerical Analysis, Computational science