Ch1Q19FiniteDiffs.m

%%  Finite differences for the advection-diffusion equation
% 
%      - D u_xx   + v u_x   =   1 
% 
%     on the domain x=[0,1],
%     with boundary conditions u(0)=u(1)=0.
%
%     Computational Science and Engineering
%     Chapter 1, Section 1.2, p 25, Question 19.
%

clear
clc
close all

% try experimenting with parameters D, v, and N.
% D=1; v=1; 
 D=0.05; v=0.2;
% D=0.05; v=2; % boundary layer
N=5+1; % mesh points 

xL=0; xR=1; h=(xR-xL)/(N-1);    % h is the spatial discretization size
x=xL:h:xR; x=x';

% True solution:
r=v/D;
C=1/(v*(exp(r)-1));
uTrue=x/v+C*(1-exp(r*x));

% diffusion matrix
e = ones(N,1); K = spdiags([e -2*e e], -1:1, N, N); K=D*K/h^2; 
%K=full(K)  % uncomment this line to see the matrix K (it will be easier if N is small)

% centered difference matrix
Del0 = spdiags([-e e], [-1 1], N, N); Del0=v*Del0/(2*h);
%Del0=full(Del0)

% upwind difference matrix
DelPlus = spdiags([-e e], 0:1, N, N); DelPlus=v*DelPlus/h;
%DelPlus=full(DelPlus)

%% Linear solve
A=Del0-K; % centered advection matrix, plus diffusion matrix
A(1,:)=0; A(end,:)=0; A(1,1)=1; A(end,end)=1; % boundary conditions
b=ones(size(x));  b(1)=0; b(end)=0;           % boundary conditions
u=A\b;    % solve with backslash (centered) 

A=DelPlus-K; % upwind advection matrix, plus diffusion matrix
A(1,:)=0; A(end,:)=0; A(1,1)=1; A(end,end)=1; % boundary conditions
U=A\b;    % solve with backslash (upwind)

%% plot results
FS='FontSize';
figure(1)
plot(x,u, x,U, '--', x, uTrue, '+', 'LineWidth', 2)
title({'- D u_{xx}   + v u_x   =   1 ',  ['   D=', num2str(D), '   v=', num2str(v), '         h=', num2str(h)]}, FS, 16)
legend('u   centered', 'U   upwind', 'true solution')
xlabel('x', FS, 16)
ylabel('u', FS, 16)

disp('Finished.')