UNIT II - FINITE DIFFERENCE AND FINITE VOLUME METHODS FOR DIFFUSION

PART A
1. Write a brief note on Finite Difference method.
Finite difference. Solution is represented by point values at mesh points. Replace
each differential term in the PDE by a corresponding finite difference approximation.
I. The original approach for CFD.
II. Easy to get high-order discretization (use high-order finite differences).
III. Doesn’t conserve mass, momentum, and energy exactly.
IV. Impractical for unstructured meshes.

2. Differentiate between explicit and implicit approaches?

Explicit Implicit
I. Relatively simple to set up and I. More complicated to set up and
program. program.
II. Δt must be very small to II. Stability can be maintained over
maintain stability much larger values of Δt
III. This can result in long computer III. This can result in less computer
running times to make running times to make
calculations over a given calculations over a given
interval of t interval of t

3. Compare the advantages & disadvantages of FDM & FVM?

FDM FVM
Basic FV control volume balance
does not limit cell shape;mass,
Advantages Easy to implement momentum, energy conserved
even on coarse grids; efficient,
iterative solvers well developed
Restricted to simple grids and
does not conserve False diffusion when simple
Disadvantages
momentum, energy, and mass numerics are used
on coarse grids

4. What do you understand by Finite volume method?

I. The FVM uses the integral conservation equation applied to control volumes
which subdivide the solution domain, and to the entire solution domain.
II. The variable values at the faces of the control volume are determined by
interpolation. False diffusion can arise depending on the choice of
interpolation scheme.
 III. The grid must be refined to reduce “smearing” of the solution as shown in
the last example.
IV. Advantages of FVM: integral conservation is exactly satisfied and the
method is not limited to grid type (structured or unstructured, Cartesian or
body-fitted).
V. Always ensure proper convergence

5. What are the main source of errors?

I. Mesh too coarse.
II. High skewness.
III. Large jumps in volume between adjacent cells.
IV. Large aspect ratios.
V. Interpolation errors at non-conformal interfaces.
VI. Inappropriate boundary layer mesh.

6. Define Discretization error?

The difference between the exact analytical solution of the partial differential
equation and the exact solution of the corresponding difference equation.
Discretization error for the difference is simply the truncation error for the
difference equation plus any errors introduced by the numerical trement of the
boundary condition.

7. Define Round-off error?

The numerical error introduced after a repetitive number of calculation in which the
computer is constantly rounding the number to some significant figure.

8. What is the concept of time marching?

In the time marching technique, the equations retain their time dependent terms even
through a steady state solution is sought. A three dimensional grid is constructed for
the whole flow field and a guess is made for the initial state of the flow variables at
all the points of the grid. A suitable guess may be 'free stream conditions
everywhere'. The solution is then stepped forward in time until convergence (or
steady state) is obtained. Time-marching can be thought of as an iterative process,
and is valid over the whole Mach number range.

9. Indicate the significance of up-wind type formulation in solving fluid flow problems.
Upwind schemes (or simply upwinding) are designed to numerically simulate more
properly the direction of the propagation of information in a flow field along the
characteristic curves. As a result, if the upwinding is carried out in a proper fashion,
the calculation of very sharp discontinuities (spread over only two grid points) with
no oscillations is possible.
10. What is Lax’s equivalence theorem?
For a PDE which approximates a well-posed, linear initial value problem, the
necessary and sufficient condition for convergence is that the PDE must be stable an
consistent.

11. What are the application of finite difference equation?

An important application of finite differences is in numerical analysis, especially
in numerical differential equations, which aim at the numerical solution
of ordinary and partial differential equations respectively. The idea is to replace the
derivatives appearing in the differential equation by finite differences that
approximate them. The resulting methods are called finite difference methods.
Common applications of the finite difference method are in computational science
and engineering disciplines, such as thermal engineering, fluid mechanics, etc.

12. Bring out the difference between FDM & FVM.

In finite difference method, the partial derivatives are replaced with a series
expansion representation, usually a Taylor series. The series is truncated usually
after one or two terms. The more term u includes, the more accurate the solution.
But it causes complexity and increase of nodes. In finite volume method, governing
equations are integrated over a volume assuming piecewise linear variation of
dependent variables. Using these integrations, you essentially balance fluxes across
the boundaries of individual volumes.

13. What are the steps for FVM Analysis?

Steps for FVM Analysis Step
1: Grid Generation Step
2: Discretization Step
3: Solution of Equations

14. Write the Disadvantage of the explicit approach.

Given Δx, Δt must be less than some limit imposed by stability constraints. In many
cases, Δt must be very small to maintain stability; this can result in long computer
running times to make calculations over a given interval of t.

15. Write the advantage of implicit approach.

Stability can be maintained over much larger values of Δt, hence
using considerably fewer time steps to make calculations over a given interval
of t. This results in less computer time.
16. What is Lax method?

………..(a)
The differencing used in the above equation, where Eq. (a) is used to represent the
time derivative, is called the Lax method.

17. Define Courant number?

(or)
What is the important stability criterion for hyperbolic equation?

C is called the Courant number. This equation says that Δt ≤ Δx/c for the numerical
solution to be stable. Moreover, Eq. (5.47) is called the Courant–Friedrichs–Lewy
condition, generally written as the CFL condition. It is an important stability criterion
for hyperbolic equations.

18. Name the important errors that commonly occur in numerical solution.
The numerical solution of this equation is influenced by two sources of error:
Discretization error, the difference between the exact analytical solution of the
partial differential equation and the exact (round-off-free) solution of the
corresponding difference equation, the discretization error is simply the truncation
error for the difference equation plus any errors introduced by the numerical
treatment of the boundary conditions.
Round-off error, the numerical error introduced after a repetitive number of
calculations in which the computer is constantly rounding the numbers to some
significant figure.

19. What do you mean by discretization?

In mathematics, discretization is the process of transferring continuous models and
equations into discrete counterparts. The grid generation process is closely related to
discretization, as it involves the process of breaking a continuous surface or volume
into discrete (finite) elements over which the discrete form of the equations are
solved.

20. What is meant by consistency?

A finite difference approximation of a PDE is consistent if the finite difference
equation approaches the PDE as the grid size approaches zero.
PART B
1. Explain the Lax-Wendroff method with suitable necessary example.
Refer CFD by John Wendt –Page no:128
2. Explain the Mac-Cormack method method with suitable necessary example.
Refer CFD by John Wendt –Page no:132
3. Derive the stability criterion for finite difference equation.
Refer CFD by John Wendt –Page no:97
4. Consider 2D convection diffusion equation

+ + = +

Discretize with central difference and Euler explicit time integration. Also give necessary and
sufficient stability condition.

Refer Mail attachment or notes

5. Derive the parabolic equation for unsteady thermal conduction

Refer Mail attachment or notes