MTH646 Quiz 4

Q1. Which of the following is a correct update formula for the Gauss-Seidel iteration in solving the finite difference equations for the Laplace PDE?

1. u_{i,j}^{(k+1)} = {1}{4} ( u_{i+1,j}^{(k)} + u_{i-1,j}^{(k)} + u_{i,j+1}^{(k+1)} + u_{i,j-1}^{(k+1)} )
2. u_{i,j}^{(k+1)} = {1}{4} ( u_{i+1,j}^{(k)} + u_{i-1,j}^{(k+1)} + u_{i,j+1}^{(k)} + u_{i,j-1}^{(k)} )
3. u_{i,j}^{(k+1)} = {1}{4} ( u_{i+1,j}^{(k+1)} + u_{i-1,j}^{(k)} + u_{i,j+1}^{(k)} + u_{i,j-1}^{(k)} )
4. u_{i,j}^{(k+1)} = {1}{4} ( u_{i+1,j}^{(k+1)} + u_{i-1,j}^{(k+1)} + u_{i,j+1}^{(k)} + u_{i,j-1}^{(k)} )

Correct answer: u_{i,j}^{(k+1)} = {1}{4} ( u_{i+1,j}^{(k)} + u_{i-1,j}^{(k)} + u_{i,j+1}^{(k+1)} + u_{i,j-1}^{(k+1)} )

Q2. The subscript (i,j) of w in the grid (see Fig. below) in Finite Difference Scheme for the solution of a PDE, are_________

1. (1,1)
2. (1,3)
3. (3,1)
4. (3,3)

Correct answer: (3,3)

Q3. A.

B.

C.

D.

Q4. The boundary conditions u(0,y)=0 and u(1,y)=y for the differential equation \frac{\partial ^ 2 u}{\partial x ^2} + \frac{\partial ^ 2 u}{\partial y ^2} = 0 shows that the length of x is

1. 1
2. 2
3. 0.5
4. 0

Correct answer: 1

Q5. The PDE \frac{\partial ^ 2 u}{\partial x ^2} + \frac{\partial ^ 2 u}{\partial y ^2} = 0 is _______________

1. Parabolic
2. Non-linear
3. Hyperbolic
4. Elliptic

Correct answer: Elliptic

Q6. Which form of the finite difference approximation is typically used for solving the Poisson equation using the Gauss-Seidel method?

1. Fourth-order central difference
2. First-order forward difference
3. First-order backward difference
4. Second-order central difference

Correct answer: Second-order central difference

Q7. Gauss-Seidel method is also known as………. Method.

1. Successive displacement
2. Jacobi’s iterative
3. Gauss Jorden
4. Gauss elimination

Correct answer: Successive displacement

Q8. A.

B.

C.

D.

Q9. The Poisson Differential Equation \frac{\partial ^ 2 u}{\partial x ^2} + \frac{\partial ^ 2 u}{\partial y ^2} = f(x,y) becomes Laplace Differential Equation when

1. f(x,y)=xe^y
2. f(x,y)=0
3. f(x,y)=1
4. f(x,y)=e^x

Correct answer: f(x,y)=0