tridiag(a,d,b,r) ... Stem(data)

3fcf3638-c000-4150-8826-fa97a0d602b1 32 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Euler implicit numerical solution for advection equation

<= The PDE solution

Rewrite in terms of Euler finite differences

... where 'u' at time step 'n' is known but at time step [n+1] is unknown. Take the unknowns to the left side of the equation.

Given appropriate boundary & initial conditions, we can write this equation for each node and solve them recursively at each time step. The resulting coefficient matrix is tridiagonal.

wave speed or free stream velocity c 0.4 :

time step Δt 0.02 :

distance increment Δx 0.008 :

total length of problem domain L 400 :

courant number ν c Δt Δx / * : 1

frequency parameter β 2 π * Δx L / * :

'i' subscript is position i 1100 range :

length of Advection p i 1100 range p i el i : for p 21 line :

'j' is time j 1 :

Define the vectors in the construct of the tridiagonal solver

i 1100 range j 1 a i j el ν - 2 / : a 11 el 0 : 21 line for for

i 1100 range j 1 d i j el 1 : for for

i 1100 range j 1 b i j el ν 2 / : b 1001 el 0 : 21 line for for

i 1100 range j 1 r i j el 100 i 5 ≤ 0 cases : for for

tridiag(a,d,b,r) ... Stem(data)

Solve a user tridiagonal vector system [adapted]. 
If you don't need more matrix manipulation, you
can solve the system directly. [Mathsoft Staff] a b c u tridiag n b rows : β 1 el b 1 el : T 1 el u 1 el : j 2 n range β j el b j el a j el β j 1 - el / c j 1 - el * - : T j el u j el a j el β j el / T j 1 - el * - : 21 line for u n el T n el β n el / : j n 1 - (1 n 2 - range u j el T j el c j el u j 1 + el * - β j el / : for u 71 line :

Spectrum stem plot for QuickPlot unicolor data Stem stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

number of collected solutions K 500 :

r soln s r : count 1 r count el r : count count 1 + : 21 line for count 2 K range r count el a d b s tridiag : s r count el : count count 1 + : 31 line for r 41 line :

CWD C:\SmathFile CurrentDirectory :

r soln CWD Advection wfile 1

A CWD Advection rfile :

data_1 p A 50 el augment : data_2 p A 125 el augment : 21 line

where Advection reflects return 1003010030 - 22 mat :

About this document . . .

It reconciliates Smath wrt the Mathcad 2nd version in

collaboration [D.M.Griffin, Jr/Steven Finch Mathsoft Staff].

tridiag(a,d,b,r) is an algorithm from the Numerical recipes

electronic book .

soln(r) collects recursively all the solutions over the K which

is the number of return trips of Advection phenomenon. It is

a vector of K nests, where each K nest is the solution over

the length of the corresponding trip. The solved system is

spooled to file, from where we recover the trip(s) of interest

for plotting. K = 200 completes a return trip. @ higher K

more can be inspected somewhere between origin and the

selected position A_"position"

]]> data_1 Stem data_2 Stem return 3 1 sys Stem(data)

Spectrum stem plot for QuickPlot unicolor data Stem stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

length of Advection p i 1100 range p i el i : for p 21 line :

t anim p A t el augment : 11 line

t 25 :

where Advection reflects return 1003010030 - 22 mat :

τ 15005 range :

t anim Stem return 2 1 sys