Heat Transfer · Coupled modes and numerical heat transfer
A one-dimensional steady conduction problem is discretised on a uniform mesh with…
Problem
A one-dimensional steady conduction problem is discretised on a uniform mesh with spacing $\Delta x=1$ and conductivity scaled so that $k=1$. The domain has five nodes labelled $i=0,1,2,3,4$. The boundary values are prescribed \[ T_0=0,\qquad T_4=0. \] The three interior unknown temperatures $T_1,T_2,T_3$ obey the second-difference balance \[ -T_{i-1}+2T_i-T_{i+1}=1,\qquad i=1,2,3. \] The right-hand side is a unit volumetric source per cell (already multiplied by $\Delta x^2/k=1$). The discrete outward boundary fluxes, consistent with the same stencil, are \[ q_L=T_1-T_0,\qquad q_R=T_3-T_4. \] 1. Write the $3\times 3$ linear system after inserting the zero boundary values. 2. Solve for $(T_1,T_2,T_3)$ exactly. 3. Evaluate $q_L$ and $q_R$ and show that $q_L+q_R$ equals the sum of the three unit sources. 4. Audit the claims (i) that the interior stencil should be written $+T_{i-1}-2T_i+T_{i+1}=1$, and (ii) that the boundary fluxes are $T_0$ and $T_4$ (hence both zero) so they cannot balance the source. Do not replace this linear three-node Poisson problem by the occupied nonlinear surface residual, and do not introduce time marching or a Fourier number.
Hint
The system is $2T_1-T_2=1$, $-T_1+2T_2-T_3=1$, $-T_2+2T_3=1$.
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