Why classical finite difference approximations fail for a singularly perturbed system of convection-diffusion equations

Thesis Overview

<p> Introduction<br>The contents of this thesis fall within the general area of numerical methods for<br>PDE, an area which has attracted the attention of prominent mathematicians<br>due to its diverse applications in numerous fields of sciences<br>1.1 Motivation<br>Imagine a river – a river flowing strongly and smoothly, liquid pollution pours into<br>the water at a certain point, which shape does the pollution stain form on the<br>surface of the river? Two physical processes operate here: the pollution diffuses<br>slowly through the water, but the dominant mechanism is the swift movement of<br>the river which rapidly convects the pollution along a one – dimensional curve on<br>the surface; diffusion gradually spreads that curve. When convection and diffusion<br>are both present in a linear differential equation and convection dominates, we<br>have a convection – diffusion problem. The simplest mathematical model of a<br>convection – diffusion problem is a two point – point boundary value problem of<br>the form,<br>(<br>ô€€€”u00(x) + a(x)u0(x) + b(x)u(x) = f(x) 0 &lt; x &lt; 1<br>u(0) = u(1) = 0;<br>(1.1)<br>where ” is a small positive parameter and a(x); b(x); f(x) are some given functions.<br>The term u00 corresponds to the diffusion and its coefficient ” is small, while the<br>expression u0 represents convection. Finally u and f(x) play the roles of a source<br>and driving term respectively.<br>Aroh Innocent Tagbo 1<br>Now, having known that the solutions of ODE’s lives in C[a; b], consider the<br>problem<br>ô€€€”u<br>00<br>(x) + u0(x) = 1 for 0 &lt; x &lt; 1 : : : : (1.2)<br>with u(0) = u(1) = 0 and 0 &lt; ” &lt; 1.<br>suppose that we formally set ” = 0; here we get<br>(<br>u0(x) = 1 for 0 &lt; x &lt; 1 : : :<br>u(0) = u(1) = 0:<br>(1.3)<br>The problem (1.3) has no solution in C[0; 1] so we infer than when ” is near<br>zero the solution of (1.3) is badly behaved. Problems like (1.3) are differential<br>equations that depend on small positive parameter ” and whose solutions (or<br>their derivatives ) approach a discontinuous limit as ” approaches zero. We say<br>that such problems are singularly perturbed where we regard ” as a perturbation<br>parameter. In more technical terms , one cannot represent the solution of a singularly<br>perturbed differential equation as an asymptotic expansion in the powers<br>of “. Moreover not every differential equation be it ODE or PDE can be solved<br>analytically and singular Perturbations arise in several branches of engineering<br>and applied mathematics, including fluid dynamics, so in investigating numerical<br>skills for tackling such problems leads to the main objective of this thesis.<br>1.2 Formulation of the problem<br>Classical Finite Difference Scheme is one of the most frequently used method for<br>numerical solution for both ordinary and partial differential equation. But on the<br>contrary, in this work we study why classical SFD scheme fails to approximate<br>a coupled system of singularly perturbed convection-diffusion. The governing<br>equations of the problem are given by<br>8&gt;&lt;<br>&gt;:<br>ô€€€”uxx ô€€€ a1(x)ux + b11(x)u + b12(x)v = f(x);<br>ô€€€vxx ô€€€ a2(x)vx + b21(x)u + b22(x)v = g(x);<br>u(0) = u(1) = v(0) = v(1) = 0:<br>(1.4)<br>where (u; v) is the solution of (1.4) above. In (1.4), we assume that<br>0 &lt; ” &lt; 1; (1.5)<br>2<br>and<br>ak(x) &gt; 0 ; bkk(x) 0 ; k = 1; 2: (1.6)<br>The convection-diffusion equation (1.4) are considered as linearised version of<br>the Navier-Stokes equation, they constitute an element of interest in the area of<br>fluid dynamics and hydro dynamics. Although the equation (1.4) may not be<br>applied directly to real applications, it is an important stage in investigation of<br>many practical applications. There is a lot of work in literature dealing with the<br>numerical solution of a single equation of (1.4) but systems of equations appear<br>relatively rare.<br>In chapter 2, we introduced the notion of the classical SFD approximation accompanied<br>with some basic definitions and results. Then we formulated the classical<br>SFD for (1.4) and showed its consistency with the continuous problem (1.4), we<br>gave an elegant proof of the existence and uniqueness of the solution of the discrete<br>operator.<br>In chapter 3 and chapter 4, stability analysis and error analysis were both investigated<br>respectively, and both turned out not to be uniform with respect to the<br>perturbation parameters (“; ). For the stability analysis, the solution blows up<br>as (“; ) goes to zero, and there will no convergence at all as (“; ) goes to zero.<br>Basically this is why the classical SFD fail to approximate (1.4), it couldn’t take<br>care of (“; ) and they found them selves in damaging positions.<br>In chapter 5, we wrote a computer program and simulate the method for several<br>cases of interest and the numerical investigations corroborated with our theoretical<br>findings.<br>3<br>4 <br></p>