Explanation: The nature of the original PDE should not be changed while solving the problem. If the problem is non-linear in PDE, to make it solvable by the implicit scheme, the difference equation should be linearized.

Explanation: The whole process is to initiate from the known values at the time-step t and move on to the required time-step t+Δ t. For this marching, the ADI scheme uses two steps. The solution does not directly move to the final step t+Δ t.

Explanation: The first step of the ADI scheme is from t to t+\(\frac{\Delta t}{2}\). The second step of the ADI scheme is from t+\(\frac{\Delta t}{2}\) to t+Δt. Here, the intermediate step t+\(\frac{\Delta t}{2}\) is extra for the process and the results here are not actually needed.

Explanation: In the first step with the time interval \(\frac{\Delta t}{2}\), the spatial derivatives are replaced using the central difference scheme. Only the x-derivative is treated implicitly. Then the resulting equations are solved using the TDMA method.

Explanation: The second step of the ADI scheme uses the time domain from t+\(\frac{\Delta t}{2}\) to t+Δt. Here, the y-derivative is treated implicitly after replacing the derivatives with central differences. The solutions are swept in the y-direction here.

Explanation: In the first step of the ADI scheme, the Thomas algorithm is used N times to solve in the x-direction. Similarly, in the y-direction, again the Thomas algorithm is used N times to get the solutions. Totally, the Thomas algorithm is used 2N times.

Explanation: The truncation error of the ADI scheme in the time direction is of order two. The ADI scheme is second-order accurate in the time direction. It uses two time-steps to move from the step t to t+Δt.

Explanation: The truncation errors of the ADI scheme in the x and the y-directions are O(Δx2) and O(Δy2). Therefore, the order of accuracy of the ADI scheme in the x and the y-directions are two and two respectively.

Explanation: The ADI scheme is useful to solve many fluid flow problems including the heat conduction and the mass diffusion problems. It is particularly suitable for the parabolic and elliptic problems.