Question

Consider a system of equations where the i^th equation is ai Φi=bi Φ(i+1)+ci Φ(i+1)+di. While solving this system using Thomas algorithm, we get Φi=Pi Φ(i+1)+Qi. What are Pi and Qi?

(a) \(P_i=\frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}};Q_i=\frac{b_i}{a_i-c_i P_{i-1}}\)

(b) \(P_i=\frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}\)

(c) \(P_i=\frac{c_i Q_{i-1}+b_i}{a_i-c_i P_{i-1}};Q_i=\frac{d_i}{a_i-c_i P_{i-1}}\)

(d) \(P_i=\frac{d_i}{a_i-c_i P_{i-1}};Q_i=\frac{c_i Q_{i-1}+b_i}{a_i-c_i P_{i-1}}\)

I have been asked this question by my college director while I was bunking the class.

The origin of the question is Discretization Aspects in portion Basic Aspects of Discretization, Grid Generation with Appropriate Transformation of Computational Fluid Dynamics

Answer 1

Right choice is (b) \(P_i=\frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}\)

Best explanation: As given,

Φi = PiΦi+1+Qi

Φi-1 = Pi-1Φi+Qi-1

The i^th equation is,

aiΦi = bi Φi+1 + ciΦi-1 + di

aiΦi = bi Φi+1 + ci(Pi-1Φi + Qi-1) + di

aiΦi – ciPi-1Φi = bi Φi+1+ciQi-1+di

Φi(ai-ci Pi-1) = biΦi+1+ci Qi-1+di

\(\Phi_i = \frac{b_i}{a_i-c_i P_{i-1}}\Phi_{i+1} + \frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}} \)

Therefore,

\(P_i = \frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{C_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}\).