Week 3 Time Discretization; Solvers Week 4 Terminology; Pseudo Code

Question 1

Q: RK method? Order of Accuracy & Type of method?

Answer 1

Q: RK method? Order of Accuracy & Type of method?

Multi-stage method
2nd order accurate globally - RK-2
3rd order accurate globally - RK-3
4th order accurate globally - RK-4
Low storage RK-3 will be used mostly in this course.

Question 2

Q: Upwind schemes?

Answer 2

Q: Upwind schemes?

Biased schemes - Discretization based on cell velocity (more weightage given to upstream point)

Question 3

Q: Overall accuracy of discretization?

Answer 3

Q: Overall accuracy of discretization?

Depends upon the discretization of both first-order & second-order derivatives.
If 1st order derivative is 2nd order accurate & 2nd order derivative is 4th order accurate - method will be 2nd order accurate

Question 4

Q: Positive Definite Matrix? Properties of PDM?

Answer 4

Q: Positive Definite Matrix? Properties of PDM?

c^* A c > 0 for all c &
c^* A c = 0 means c=0
Has Real & +ve Eigen values
Symmetric PDM has Unique Inverse

Question 5

Q: What is Thomas Algorithm for? Psuedo code?

Answer 5

Q: What is Thomas Algorithm for? Psuedo code?

for solving tri-diagonal system of equations
a specific case of GE for Tri-diagonal system of equations

Question 6

Q: Grid Type? Collocated & Staggered

Answer 6

Week 4

Q: Grid Type? Collocated & Staggered

Collocated - both p & velocity components are solved at nodes/ cell boundaries
Staggered - p at cell centre & velocity components are solved at cell boundaries

Question 7

Q: Types of BC?

Answer 7

Q: Types of BC?

Dirichlet BC - value - like no-slip
Neumann BC - gradient/ derivative - like heat flux
Periodic BC - outlet used as inlet - flow invariant, reduces the computational requirement

Question 8

Q: Direct & Iterative Methods

Answer 8

Q: Direct & Iterative Methods?

Direct Methods -
Iterative Methods - Jacobi, Gauss-Seidel (GS), Successive Over-Relaxation (SOR)
Jacobi - (n+1)th level using n(th) values
GS - use best possible updates
SOR - weighted avg. of nth & (n+1)th value

Question 9

Q: Psuedo-code for GS

Answer 9

Q: Psuedo-code for GS

allot arrays
save coefficients
write the iter equation
update Boundary values
calculate L2 norm & check for convergence