ch2: Finite Differences

Question 1

A Taylor series is

Answer 1

Question 2

express the first derivative of the following is

Answer 2

Question 3

Taylor’s series expansion can be used to

Answer 3

express the derivatives

Question 4

The expressions on the RHS of (3) and (4) that are used to express

Answer 4

the space derivatives, are called the forward and backward finite differences.

Question 5

The difference between the derivative and its finite difference expression is shown in the following diagram.

Answer 5

Question 6

what is the tangent and secant line

Answer 6

The straight line that”just touches” the curve at a point is the tangent, while the line that intersects two points on the curve, is the secant line.

Question 7

what does the tangent and secant line represent

Answer 7

the slope of atangent line represents the derivative, the slope of a secant line represents the finite difference.

Question 8

as /-\x approaches zero

Answer 8

the slope of the secant line approaches the slope of the tangent line. Then, the finite difference will be the true derivative.

Question 9

The fundamental equations governing the atmospheric motion consist of

Answer 9

non‐ linear partial differential equations, which do not have analytical solutions and are solved using numerical methods.

Question 10

The fundamental equations governing the atmospheric motion consist of non‐ linear partial differential equations, which do not have analytical solutions and are solved using numerical methods.

These equations include

Answer 10

spatial derivatives at a fixed point, which can be approximated based on Taylor’s expansion of the variable about that point

Question 11

from the taylor’s series expansion (1 & 2), ………………………….. may be formulated for the ………………………………………………..

Answer 11

three different expressions

approximation of the first derivative of the function u at x

Question 12

The forward and backward difference expressions can be obtained from (1) & (2):

Answer 12

Question 13

The centered difference expression can be obtained by

Answer 13

Question 14

Thus the error (truncation error)

Answer 14

Question 15

Thus, the first derivative can be approximated by forward, backward and center
difference formulae as:

Answer 15

Question 16

The second order ……………….. accurate second derivative of ………. may be easily obtained by

Answer 16

Question 17

The Laplacian operator

Answer 17

The Laplacian operator appears in many diagnostic and prognostic equations in meteorology. The application of its finite analog is found to be very practical for the solution of many problems.

Question 18

The Laplace’s equation takes the form:

Answer 18

Question 19

The 3D and 2D Laplacian operator in Cartesian coordinate system, can be expressed as:

Answer 19

Question 20

let ………………. be any scalar function of …………………………… the finite difference analog of ………………………………………….. will be evaluated with reference to ……………………. as follows

Answer 20

Question 21

use taylor series to expand a function of two variables

Answer 21

Question 22

there is ……………………………… which also gives second order accurate results this uses …………………………………….

Answer 22

Question 23

using …………………………. is slightly more accurate than ………………………………………….

Answer 23

Using a 5‐point diamond stencil is slightly more accurate than a 5‐point square stencil

Question 24

Using a 5‐point diamond stencil is slightly more accurate than a 5‐point square
stencil. this is because

Answer 24

the distance from the center point (I,J) to the other points is less in the diamond stencil than the corresponding distance in the square stencil.

Question 25

Jacobian is also

Answer 25

a common operator encountered during the solution of many geophysical problems. It appears mostly in the non‐linear advective terms.

Question 26

Jacobian is also a common operator encountered during the solution of many geophysical problems. It appears mostly in the non‐linear advective terms.
For example

Answer 26