Develop an artificial selective damping stencil for a high-order scheme involving 17 mesh points

7.3. In some applications, a symmetric stencil with regular size mesh may not be the most appropriate. Consider the seven-point stencil as shown,

*—— t—- *—– t—- ¥—– Ф—- ¥—– t—- *—– t—- *

Develop a set of damping coefficients for the stencil.

7.4. In two-dimensional problems artificial selective damping is added both in the x and y directions. For instance, the continuity equation with damping terms added

may be written as follows:

(a) Plot contours of constant damping rate in the a Ax – в Ay plane for – n <

a Ax, в Ay < n (assume Ax = Ay).

(b) What is the damping rate for a wave propagating at an angle of 30 degrees to the positive x-axis; the total wave number к Ax = 2.5, where к = (a2 +

в 2)1/2.

7.5. Compute the solution of the convective wave equation

d U d U

T- + T – = 0

d t dx

on a uniform mesh with Ax = 1 and the following initial condition,

t = 0, u = [2 + cos(ax)] exp[- (ln 2) (x/10)2]

Consider two cases.

(i) a = 1.7

(ii) a = 4.6

Results to be reported are the spatial distributions of u at t = 400 and t = 800. The exact solution to (i) is

2

 

x – t

 

u(x, t) = [2 + cosa(x – t)] exp

 

(ln2)

 

10

 

The exact (aliased) solution to (ii) is

u(x, t) = [2 + cos[(2n – a)(x -1)]] exp

 

x – t

“io”

 

(ln2)

 

8

 

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