Question

Which of these equations is the discretized form of the transient term using the first-order implicit Euler scheme?

(a) (frac{( ho_Cphi_C)^t-( ho_Cphi_C)^{t+Delta t}}{Delta t} V_C+L(phi_C^t))

(b) (frac{( ho_Cphi_C)^t-( ho_Cphi_C)^{t-Delta t}}{Delta t} V_C+L(phi_C^t))

(c) (frac{( ho_Cphi_C)^t+( ho_Cphi_C)^{t+Delta t}}{Delta t} V_C+L(phi_C^t))

(d) (frac{( ho_Cphi_C)^t+( ho_Cphi_C)^{t-Delta t}}{Delta t} V_C+L(phi_C^t))

Answer 1

Right choice is (b) (frac{( ho_Cphi_C)^t-( ho_Cphi_C)^{t-Delta t}}{Delta t} V_C+L(phi_C^t))

The explanation is: The first-order implicit Euler scheme gives its terms using the older terms. Before using this scheme, the terms are

(frac{V_C( ho_C phi_C )^{t+frac{Delta t}{2}}}{Delta t}-frac{V_C( ho_Cphi_C)^{t-frac{Delta t}{2}}}{Delta t}+L(phi_C^t))

When the scheme is applied to these equations,

(frac{( ho_Cphi_C)^t-( ho_Cphi_C)^{t-Delta t}}{Delta t} V_C+L(phi_C^t)).