Tridiagonal Matrix Algorithm (TDMI)

A second order linear differential equation was solved by the analytical and numerical approach. Finite difference method was used to discretize the differential equation. A second order central difference scheme with four different meshes (N = 6, 11, 21, 41) was used. The discretized equations were solved by tridiagonal matrix inversion (TDMI) algorithm. A Dirichlet-type boundary condition was used at the boundary nodes. Figure 1(a) shows the analytical solution where is phi plotted as a function of x. Figure 1 shows the numerical solution (N = 11 case only) and compared with the analytical solution.

MATLAB Code

%% Tridiagonal Matrix Algorithm

% Arif Hossain - 01/19/2017

clc; clear all; close all;

mesh = [6 11 21 41];

 for j = 1:length(mesh)

  % Analytic Calculation

c1 = 2-exp(-1);

c2 = -1;

x_any = linspace (0,1,100);

phi_any = exp(-x_any)+c1*x_any+c2;

  % Numerical calculation   

N = mesh(j);

L = 1;

delX = L/(N-1);

x = (0:delX:L);

D = [1, -2*ones(1,N-2), 1];

C = [0, 1*ones(1,N-2)];

A = [1*ones(1,N-2), 0];

B(1) = 0;B(N) = 1;

phi(1) = 0;

for i = 2:N-1

            B(i) = (delX.^2)*exp(-i*delX);   

end

for i = 2:N-1

            xmult = A(i-1)/D(i-1);

            D(i) = D(i)-xmult*C(i-1);

            B(i) = B(i)-xmult*B(i-1);

end

phi(N) = B(N)/D(N);

for i = N-1:-1:1

            phi(i) = (B(i)-C(i)*phi(i+1))/D(i);

end

  % Figure

figure()

plot(x_any,phi_any,'-r','linewidth',2)

hold on

plot(x,phi,'sb','linewidth',2)

set(gca,'fontsize',14,'fontweight','bold')

xlabel('x');ylabel('\phi(x)');grid on

legend('Analytic sol^n','Numeric Sol^n (TDMA)','location','southeast')

  % Error calculation

for i = 1:N

            phi_any(i) = exp(-x(i))+c1*x(i)+c2;

            err(i,j) =  abs((phi_any(i)-phi(i))/phi_any(i))*100;

end

figure(2)

plot(x,err(:,j),'--s','linewidth',2)

set(gca,'fontsize',14,'fontweight','bold')

xlabel('x');ylabel('E(x)');grid on

hold on

end