Jordan Wall
Studies in CFD
Fall 2012
Homework 1
1.- Describe and discuss the features and differences among
the various types of errors encountered in numerical
computation: truncation errors, round-off errors and
discretization error.
Truncation error is a result of approximating the PDE by the
finite difference expression algebraic equations used to
represent the derivative terms in the PDE. The algebraic
equations are derived from rearranging the taylor series
expansion of the expression for the grid points: u(x0 + x,y0),
u(x0 - x,y0), u(x0,y0 + y0), etc. Here the truncation error is the
difference between the partial derivative and its finite
difference representation. TE = PDE – FDE. The order of the
truncation error is represented by the lowest order derivative
from the remainder terms in the taylor series expansion of the
PDE. Truncation error is unavoidable when trying to perform
finite calculations of an infinite series; however truncation error
can be reduced by using higher order TE terms i.e (O(x)2
instead of O(x)).
Round off errors result from a computer rounding to a finite
number of digits when performing arithmetic operations on the
finite difference expressions. Round off errors may increase as
grid points increase and subsequently the arithmetic operations
increase.
Discretization Error is the error in the solution to the PDE caused
by replacing the continuous problem by a discrete one.
Discretization error is the sum of the truncation error plus any
error introduced by the treatment of the boundary conditions.
2.- Describe and explain the meaning of numerical stability.
The concept of numerical stability is one which ensures that
errors (round off or truncation) do not grow over time as
marching steps are executed. Numerical stability is a concept
applicable only to marching problems (initial value problems).
Fourier analysis can be used to determine the maximum step
size required to ensure stability.
3.- Describe and explain what is meant for the conservative
property of a PDE
In fluid mechanics and heat transfer the physical laws such as
conservation of mass, momentum, and energy must be
maintained. To derive the PDEs that represent these physical
processes you must start with the control volume form of the
conservation statement. The PDEs once constructed, preserve
the integral conservation relations of the continuum. This is
known as the conservative property of a PDE. In order for the
FDE to maintain this integral relation, and hence have the
conservative property, the FDE must maintain the discretized
version of the conservation statement exactly (except for roundoff errors) for any mesh size over an arbitrary finite region
containing any number of grid points.
4.- Describe and explain the features and differences that exist
between finite difference and finite volume methods used in
the discretization of PDEs.
The finite volume and finite difference methods begin with
establishing a grid in the medium of observation but then they
diverge from there. The finite difference method then solves the
continuous PDE by replacing the partial derivatives with finite
difference representations at each grid point. The algebraic
expressions for the PDE are then solved using one of various
known methods.
After a grid is established in the finite volume method, the next
step is to fix the boundaries of the control volumes. The integral
form of the conservation statement is then applied to the
control volume, and the numerical equations are then derived.
It is important to note that the same answer can be derived
from both methods, but they will not always give identical
results. Also it seems that discretization based on the finite
volume method would have the conservative property, due to
the conservation laws being applied over control volumes in
space.