ECP1 Software Manual

7/30/2019 ECP1 Software Manual

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V. ZALIZNIAK

ESSENTIALS OF COMPUTATIONAL PHYSICS

Part 1: An Introduction to finite-difference methods

The manual contains a short description of programs accompanying this text. These programs can be used inComputational physics and Numerical methods courses as well as in students research projects. Additionalinformation on how to use the accompanying programs can be found in examples. Although great care is takento eliminate all program errors, the author does not take any responsibility for the consequences of application ofthese programs.

This manual consists of two parts: A short description of mathematical models and finite-difference schemesand A short description of MATLAB functions.

V. Zalizniak, 2005


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A short description of mathematical models and finite-difference schemes

Heat transfer

One-dimensional heat transfer (uniform medim).

Equation:2

2, 0 , 0

u ut T x l

t x

=

,

where u temperature, =a/c - thermal diffusion coefficient, a thermal conductivity coefficient, c specific

heat, - density.

Initial conditions: u(x,0)=g(x).

Bundary conditions:

or1(0, ) ( )u t f t = 1(0, ) ( )u

a t fx

=

t ,

or2( , ) ( )u l t f t = 2( , ) ( )

u

a l t f t x

= .

Finite-difference scheme:

1 1 111 1 1

2

2 2(1 )

0 1, 0,1,...; 1,..., 1

k k k k k kk knn n n n n n u u u u u uu u

h

k n N

+ + ++

+ + + + = +

= =

12

n

h , (1)

One-dimensional heat transfer (nonuniform medim).

Equation:

( ) ( ) ( ) , 0 , 0u u

x c x a x t T x l t x x

= .

Initial conditions: u(x,0)=g(x).

Bundary conditions:

or1(0, ) ( )u t f t = 1(0, ) ( )u

a t fx

=

t ,

or2( , ) ( )u l t f t = 2( , ) ( )u

a l t f t x

=

.



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3

1 1 1 111 1

1/2 1/22 2

1 11/2 1/22 2

( )

(1 )

k k k kk kn nnn n n

n n n

k k k kn n n n

n n

u u u uu uc a a

h h

u u u ua a

h h

+ + + +++

+

+ +

= +

(2)

Nonlinear equation:

( )( ) ( ) ( ) ,

0 , 0

uu c u u a u

t x

t T x l

=

x


1 1 1 111 1

1/2 1/22

k k k kk kn nnn nk k k k n

n n n nk

u u u uu uc a a

h h

+ + + +++

+

=

2, (3)

where

( )1 11/2 2

( )

( )

( )

k kn n

k kn n

k kn nn

u

c c u

a a u u

++

=

=

= +k

.

Two-dimensional heat transfer (uniform medim).

Equation:2 2

2 2( , , ) ,

0 , 0 , 0

s

x y

u u uf x y t

t x y

t T x l y l

= + +

The explicit splitting-up scheme:

, ,, ( , , ),

1,..., 1; ,...,

kn m n m k

x n m s n m k

s e

v uu f x y t

n N m m m

= +

= =

(4)1

, ,, ,

,..., ; 1,..., 1

kn m n m

y n m

s e

u vv

n n n m M

+ =

= =

where

( )

( )

, 1, ,2

, , 1 , ,2

12

12

k k k k x n m n m n m n m

k k k k y n m n m n m n m

u u u u h

u u u u h

+

+

= +

= +

1,

1

.

Papameters ns, ne, ms, medepend on boundary conditions:

,1 , 1 0

0 , 2 0s

boundary conditions of the st kindat xn

boundary conditions of the nd kind at x

==

=


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4

1 , 1

, 2

x

ex

N boundary conditions of the st kindat x ln

N boundary conditions of the nd kindat x l

== =

,1 , 1 0

0 , 2 0sboundary conditions of the st kindat y

mboundary conditions of the nd kindat y

== =

.1 , 1

, 2

y

ey

M boundary conditions of the st kindat y lm

M boundary conditions of the nd kindat y l

== =

One-dimensional heat transfer with phase transition (Stefan problem)

Equation:

( , ) , 0 , 0H u

a x t x l t t x x

= >

,

,( )

( ) ,

s p

l l s p

c u u u H u

c u c c u L u u


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5

Propagation of acoustic waves

Second order differential equation:2 2

22 2

, 0 , 0u u

c x l t t x

=

T,

where u velocity or pressure andc speed of sound.

Initial conditions: 1 2( , 0) ( ) , ( , 0) ( )u

u x g x x g x t

= =

,

Boundary conditions:

1 2(0, ) (0, ) ( )uu t t f t x + =

1,

3 4( , ) ( , ) ( )u

u l t l t f t x

+ = 2

.

Finite difference scheme cross:

1 11 12

2

22

1 , ..., 1 , 2 , 3 , ...

k k kk k kn n nn n n u u uu u u c

hn N k

+ + + +

=

= =

2 , (6)

System of differential equations:

, 0 , 0x l t T t x

+ =

s s

A o ,

where

, , u

p =

s2

0 1/

0c

= A

0

0

= o

u velocity, p pressure, - density andc speed of sound.

Initial conditions: ,1 2( , 0) ( ) , ( , 0) ( )u x g x p x g x = =


,1 2(0, ) (0, ) ( )u t p t f t + = 1

2 .3 4( , ) ( , ) ( )u l t p l t f t + =

Lax-Wendroff scheme:

( ) ( ) ( )2

1 21 1 1

12

2 21 , ... , 1

k k k k k k k n n n n n n n h h

n N

+

+ + = + +

=

s s A s s A s s s 1, (7)

or in expanded form

( ) ( )1 21 1 11

22 2

k k k k k k k n n n n n n n

ru u p p r u u u

c+

+ + = + + 1 ,

( ) ( )1 21 1 11

22 2

1 , ... , 1

k k k k k k k n n n n n n n

rcp p u u r p p p

n N

+

+ + = + +

=

1.

Nondissipative scheme of P. Roe [2]:


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6

1 1

1 1 11 02

k k k k k k n n n n n n R R R R R Rc

h

+ + +

= ,

111 1 11 0

2

k k k k k kn n n n n n S S S S S S c

h

++ + +

+ =

, (8)

1 , ... , 1n N=

, ,k k knn nR p c u = + k k kn n nS p c u =

= 0.5( )k k kn n nS= +, p R .0.5( )/k k kn n nu R S c

Godunov scheme:

( )11/2 1/2 3/2 1/2

3/2 1/2 1/222 2

k k k k n n n n k k k

n n n

u u p p cu u u

h h

+

+ + + + +

= + + ,

(9)

( )11/2 1/2 3/2 1/22

3/2 1/2 1/222 21,..., 2 ; 0,1,...

k k k k n n n n k k k

n n n

p p u u cc p p

h hn N k

+

+ + + + +

= + +

= =

p.

PEM scheme [4]:

1 11 1

1 11/2 1/2 1

( ) ( )1

( ) ( )

k kk k nn

k kn n nn

u u f u f p p h

u

f p f p

+ ++ +

+ +

+ + +

= A

, (10)

where

( )

( ) ( )

1 1 1

2 31 11/2 1/2

1

2

2 2

k k kn n n

k k k k k n nn n

u R Sc

a aa u R S R S

c c

+ + +

+ +

= =

+ + 1n,

( )

( ) ( )

1

2 31 11/2 1/2

1( ) ( ) ( )

2

2 2

k k kn n n

k k k k k n nn n

f u f R f Sc

b bb u R S R S

c c

+

+ +

= =

+ + 1n

,

( )

( ) ( )

1 1 1

2 31 11/2 1/2

1

2

2 2

k k kn n n

k k k k k n nn n

p R S

a aa p R S R S

+ + +

+ +

= + =

+ + + 1n,

( )

( ) ( )

1

2 31 11/2 1/2

1( ) ( ) ( )

2

2 2

k k kn n n

k k k k k n nn n

f p f R f S

b bb p R S R S

+

+ +

= + =

+ + + 1n

n nR p c u = + k k kn n nS p c u =

,

, .k k kn

Substitution of a grid solution of the form

( )exp ( )kn ku u i t kx = n (11)


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7

into finite-difference scheme produces dispersion relation =(k,,h), where - circular frequency and k

wave number. Then this solution dissipates as exp(-(Im )t)=exp(-t), - dissipation coefficient, and the group

velocity of harmonic wave (11) is defined as

Re gd

cd k

= .

Gasdynamics

Equations of motion in the form of conservation laws:( )

t x

+ =

s f s

o ,

where s andf(s) have the form

( )

1

2

2132

s

s u

s e u

= = +

s ,

( ) ( )

21

2 2

2 2

213 2 3 12

( ) //

u sf

f u p s sf s s p s u e u p

= = + = + + + +

f s1 p .

Equation of state:

( )1

pe

=

,

then

( ) ( )22

31

1 12

sp e s

s

= = .

Here u velocity, p pressure, - density ande specific internal energy, -adiabatic exponent.

Two-stage Lax-Wendroff scheme with artificial viscosity:

( ) ( ) ( )( )1/2 1 11/ 21

2 20 , ... , 1

k kk k k k n n n n n h

n N

++ ++

= +

=

s s s f s f s,

(12

( ) ( )( ) ( )1/2 1/21 1 11/2 1/2 21 , ... , 1

k kkk k k k k n n n n n n nh

n N

+ ++

+ + = + +

=

s s f s f s s s s,

where - constant parameter of artificial viscosity.


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Godunov scheme:

( ) ( )1 11/2 1/2

0 , ... , 1

k k k kn nn n

k h

n N

+

++ +

+

=

s s f s f so=

. (13)

Values of

( )( )1,kn n kx t+=s s w

are determined from the solution of Riemann problem with the initial condition

, (14)( )1/2

1/ 2

,

, ,

knn

k knn

x x

x t x x

+

= >

w

ww

where

.u

p

=

w

A state version of the method WAF (Weighted Average Flux) [3] plus artificial viscosity term:

( ) ( )(

111/2 1/2

1 2

0 , ... , 1.

k k k kn nn n k k k

n n nk h

n N

+

++ +

+

= +

=

s s f s f ss s s

)1, (15)

where

,( )k kn n=s s w

/2

/2

1( , 0.5 )

n

n

x h

kn k

x h

x t dx h

+

= +w w

k

andw(x,t) the solution of Riemann problem with the initial condition (14).

A TVD version of the method WAF [3]:

( ) ( )

1

11/2 1/2

0 , ... , 1.

k k k k

n nn nk h

n N

+

++ +

=

=

s s f s f s

o

,

where is modified with the use of a scalar limiter function (q,r), qis the ratio of the upwind change to the

local change for some gasdynamical variable, ris the Courant number. In [3] q= orq=e, here this method is

modified by introducing q= (,u,p) and, therefore, =(

knw

1,2,3).


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9

Stationary equations

Two-dimensional Helmgoltz equation with constant coefficients:

( )

22

2 2, ,

{0 , 0 }x y

uua bu

x y

D x l y l

+ + = =

f x y

)

M

with Dirichlet boundary conditions

and ,1( ) g (u 0,y = y 3( , )=g ( )xu l y y

and .2( ,0)=g ( )u x x 4( , )=g ( )yu x l x

Five-point approximation:

, (16)1 -1 2 -1 3 2 +1, 1 +1n,m n ,m n,m n m n,m n,m c u c u c u c u c u f + + + =

,1,..., 1; 1,..., 1n N m M = =

,( ) ( )0, 1 3, , =0,...,m m N,m m u g y u g y m = =

,( ) ( ),0 2 , 4, , =1,..., -1n n n M n u g x u g x n N = =

where

( )1 2 3 1 22 2, , 2y x

a ac c c c c

h h= = = + b .

Two-dimensional Helmgoltz equation with variable coefficients:

( ) ( ) ( ) ( ), , ,

{0 , 0 }x y

u ua x y a x y b x y u f x y

x x y y

D x l y l

+ + = =

, ,

,m

(17)


and ,1(0, )= ( )u y g y 3( , )= ( )xu l y g y

and .2( ,0)= ( )u x g x 4( , )= ( )yu x l g x

Five-point approximation:

, (18)(1) (2) (3) (2) (1)

, 1 -1, , , , +1, , , +1, 1 1,

n m- n m n m n m n m n m n m n m n n m- n- m

c u c u c u c u c u f + + + =

.=1,..., -1; =1,..., -1n N m M

where, +1/2(1)

, 2

n mn m

y

ac

h= ,

1/2,(2), 2

n mn m

x

ac

h

+= ,

1/2, 1/2, , 1/2 , 1/2(3), ,2 2

n m n m n m n m n m n m

x y

a a a a c b

h h

+ + + += + ,

( )11/2, 2= ,n m n x m a a x h y + + ,

( )1, 1/2 2= ,n m n m y a a x y + + h ,


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10

,( ), = ,n m n m b b x y

.=1,..., -1; =1,..., -1n N m M

Iterative method SOR for solving system of difference equations (18) has the form:

( )

( )

(1) ( +1) (2) ( +1) ( ) ( )( +1) (2) (1), ,, -1 , -1 1, 1, 1, , +1(3)

,

( ),1

k k k k kkn m n m n m n m n m n m n- m n - m n m n m

n m

kk n m

u c u c u c u c u f c

u

+= + + + +

, ,

M

N

,

,=1, ... , -1 ; =1, ... , -1 ; =0 , 1 , ...n N m M k

,( 1) ( 1)

1 30, ,( ) , ( ) , 0 , ... ,k k

m mm N mu g y u g y m + +

= = =

,( 1) ( 1)

2 4,0 ,( ) , ( ) , 0 , ... ,k k

n nn n Mu g x u g x n + +

= = =

where kis the relaxation parameter, which is computed on every iteration with the use of the power method.

Iterative method of Concus and Golub [1] for solving equation (17) has the form:

( )( ) ( )2 2 ( 1)

( 1) ( )12

1

kk k

nn

vv f v f

x

+

+

=

= +

x 2 x , (19)

k=0, 1, .

where

( )v u a= x .

( ) ( )22

1 21

( ) ( )1( )( ) nn

a bfaxa

=

=

x xxxx

,

( )2( )

( )

ff

a=

xx

x,

( ) ( )( )1 max ( ) min ( )2 D Da a = +x xx x .To solve equation (19) five-point approximation of the form (16) can be used.

Two-dimensional nonlinear Poisson equation:

( ) ( ) ( ), , , , , ,

{0 , 0 }x y

u ua u x y a u x y f x y

x x y y

D x l y l

+ =

=

(20)


and ,1(0, )= ( )u y g y 3( , )= ( )xu l y g y

and .2( ,0)= ( )u x g x 4( , )= ( )yu x l g x

Method of time development based on ADI scheme for parabolic PDE:

( ) ( )

( ) ( )

1/2, 1/2,, ,1, , , 1,2 2

, 1/2 , 1/2, 1 , , , 12 2

2 2

1( , )

22 2

k kkn m n m n m n m

n m n m n m n m k x x

k kn m n m k k k k

n m n m n m n m n m y y

a aw vw w w w

h h

a av v v v f x y

h h

+ +

+ +

= +

, (21)

,1,..., 1; 1,..., 1; 0,1,...n N m N k =

=

=


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11

( ) ( )

( ) ( )

11/2, 1/2,, ,

1, , , 1,2 2

, 1/2 , 1/21 1 1 1, ,, 1 , 12 2

2 2

1( , )

22 2

k kkn m n m n m n m

n m n m n m n m k x x

k kn m n m k k k k

n m n m n m n m n m y y

a av ww w w w

h h

a av v v v f x y

h h

++

+

+ + + + ++

= +

,

.1,..., 1; 1,..., 1; 0,1,...n N m N k = = =

,0, g( , )n m n m v x y=

,1 11 30, ,g ( ) , g ( ) , 0 , ... ,k k

m mm N mv y v y m + +

= = = M

N ,1 12 4,0 ,g ( ) , g ( ) , 0 , ... ,k kn nn n Mv x v x n + += = =

where

( )( )1 1, , 1/21/2, 2 , ,k k k

n m n m n m n ma a v v x + ++ = + y ,

( )( )1 , 1 , 1/2, 1/2 2 , ,k k k

n m n m n m n ma a v v x y + ++ = + .

Viscous incompressible fluid

Two dimensional Navier-Stokes equations in variables stream function-vorticity:

2 2

2 2

1( , , )

Resu v f x y

t x y x x

+ + = + +

t ,

0,2 0,1( , , 0)

g gx y

y x

= ,

, (22)( ) { }, = 0 , 0 ,x yx y D x l y l t 0

2 2

2 2( , , )x y t

x y

+ =

,

u y

v x

= ,

where u, v components of velocity, Re Reynolds number, - stream function, - vorticity.


( ) 10,

0

( , )y

u

v g y t

= ,

( ) 3,

0

( , )xl y

u

v g y t

= ,

( )

2

,0

( , )

0x

u g x t

v

= ,

( )

4

,

( , )

0yx l

u g x t

v

= ,

or

( )10,

0

( , )y

x g y t

= ,

( ) 3,

0

( , )xl y

x g y t

= ,


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12

( )2,0

0

( , )x

y g x t

= ,

( ) 4,

0

( , )yx l

y g x t

= .

The explicit splitting-up scheme:

0,0

mds

dt= ,

,0

N mds

dt= ,

, 1, 1, 1, ,, 2

2

2 Re

n m n m n m n m n m n m kn m

x x

ds s s s s s u

dt h h

+ + += +

1,

,

tkttk+kn=1, , N-1; m=1, , M-1,

.( ), ,kn m k n m s t =

(23)

,00

ndq

dt= ,

,0

n Mdq

dt= ,

, , 1 , 1 , 1 ,, 2

2

2 Re

n m n m n m n m n m n m kn m

y y

dq q q q q s v

dt h h

+ + += +

, 1

,

tkttk+k n=1, , N-1 ; m=1, , M-1,

.( ) ( ), ,n m k n m k k q t s t = +

The successive solution of these systems gives

.( )1, ,kn m n m k k q t

+= +

After that, find the stream function from the finite-difference equation1 1 1 1 1 1, ,+1, -1, , +1 , -1 1

,2 2

2 2k k k k k k n m n m n m n m n m n m kn m

x yh h

+ + + + + +

+ + +

+ = , (24)

1,..., 1; 1,..., 1n N m M = =

and then compute the components of velocity:1 1

, 1 , 11, 2

k kn m n m k

n my

uh

+ ++ +

= , (25)

1 11, 1,1

, 2

k kn m n m k

n mx

vh

+ ++ +

= ,

.1,..., 1; 1,..., 1n N m M = =

Systems of ordinary differential equations (23) are solved with the use of 3rd

order Runge-Kutta scheme, whichis defined by the following Butcher table:

0

1/2 1/2

3/4 0 3/4

2/9 1/3 4/9


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A short description of MATLAB functions

Main function Examples and auxiliary functions

heat_1d_un example4_1,f1_e41, f2_e41, fs_e41

heat_1d_nun example4_2,

f1_e42, f2_e42, fs_e42heat_2d_es example4_3,

g1_e43, g2_e43, g3_e43, g4_e43,fs_e43

stefan_1d example4_4,f1_e44, f2_e44, fs_e44

properties cross example5_1,

f1_e51, f2_e51lw_acoustics example5_2,

f1_e51, f2_e51roe example5_3,

f1_e51, f2_e51godunov_ acoustics example5_4,

f1_e51, f2_e51pem example5_5,

f1_e51, f2_e51lw_gasdynamics example6_1, example6_2godunov_gasdynamics example6_3waf_gasdynamics example6_4waf_tvd_gasdynamics example6_5,

limiter_mina, limiter_supera,limiter_ultraa, limiter_vanAlbada,limiter_vanLeer, limiter_Lin

riemann

helmgoltz_2d_sor example7_1,g1_e71, g2_e71, g3_e71, g4_e71,a_e71, b_e71, f_e71

helmgoltz_2d_fft example7_2,g1_e71, g2_e71, g3_e71, g4_e71,f_e71

helmgoltz_2d_cg example7_3,g1_e73, g2_e73, g3_e73, g4_e73,a_e73, b_e73, f_e73

poisson_2d_td example7_4,g1_e74, g2_e74, g3_e74, g4_e74,a_e74, f_e74

ns_2d_vsf example8_1,

g1_e81, g2_e81, g3_e81, g4_e81


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14

function [uu]=heat_1d_un(ud,kappa,a,nx,h,time,tau,b,gamma,bl,br,fbl,fbr,fs)

Purpose Computation of one time step fromscheme (1)

Input parameters ud Temperature at the instant of time tk kappa Thermal diffusion coefficient ()

a Thermal conductivity coefficient

nx Number of grid nodesh Space step

time Instant of time tk+1tau Time stepb Parameter/h

2

gamma Parameter=0 explicit scheme>0 implicit scheme

bl,br Type of boundary condition at x=0x=l,respectively.

=0 boundary condition of the 1st kind=1 boundary condition of the 2ndkind

fbl, fbr Names of m-files in which thedependence of temperature or heat flux

on time is defined at x=0 and x=l,respectively.

fs Name of m-file in which the dependenceof external source on space and time isdefined.

Output parameters uu Temperature at the instant of time tk+1


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15

function [uu]=heat_1d_nun(ud,roc,a,nx,h,time,tau,b,gamma,bl,br,fbl,fbr,fs)


Input parameters ud Temperature at the instant of time tk roc Array of values of c in the nodes of

main grid.

a Array of values of a in the nodes ofauxiliary grid.

nx Number of grid nodes.h Space step.time Instant of time tk+1tau Time stepgamma Parameter

=0 explicit scheme>0 implicit scheme

bl,br Type of boundary condition at x=0x=l,respectively.=0 boundary condition of the 1st kind=1 boundary condition of the 2ndkind.



fs Name of m-file in which the dependenceof external source on space and time is

defined.


Note This function can be used for solving of

nonlinear equation (scheme (3)). Oneneeds only to recalculate the values of

roc, a andtauon every time step.


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function [uu]=heat_2d_es(ud,kappa,a,nx,ny,hx,hy,tau,time,bxl,bxr,fbxl,fbxr,byl,byr,fbyl,fbyr,fs)


Input parameters ud Temperature at the instant of time tk kappa Thermal diffusion coefficient ()

a Thermal conductivity coefficient

nx, ny Number of grid nodes along x y,respectively.

hx, hy Space steps along xy, respectively.time Instant of time tk+1tau Time stepbxl,bxr Type of boundary condition at x=0

x=lx, respectively.=0 boundary condition of the 1st kind=1 boundary condition of the 2ndkind.

fbxl, fbxr Names of m-files in which thedependence of temperature or heat flux

on time andy is defined at x=0 andx=lx,respectively.

byl,byr Type of boundary condition at y=0 y=ly, respectively.=0 boundary condition of the 1st kind=1 boundary condition of the 2ndkind.

fbyl, fbyr Names of m-files in which thedependence of temperature or heat fluxon time andx is defined at y=0 andy=ly,respectively.

fs Name of m-file in which the dependenceof external source on space and time is

defined.



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function [entu,uu,sf]=stefan_1d(entd,ud,ro,cs,cl,as,al,lh,up,nx,h,tau,time,bl,br,fbl,fbr,fs)


Input parameters ud, entd Temperature and enthalpy at the instantof time tk

ro Densitycs, cl Specific heat of solid and liquid phase,

respectively.

as, al Thermal conductivity coefficient of solidand liquid phase, respectively.

lh Latent heat of melting.up Temperature of melting.nx Number of grid nodes.h Space step.time Instant of time tk+1

tau Time step.bl,br Type of boundary condition at x=0x=l,

respectively.=0 boundary condition of the 1st kind=1 boundary condition of the 2ndkind.


on time is defined at x=0 and x=l,

respectively.fs Name of m-file in which the dependence

of external source on space and time isdefined.

Output parameters uu, entu Temperature and enthalpy at the instantof time tk+1

sf Local fraction of solid phase at the instantof time tk+1


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18

function [dis,cg]=properties(kh,r,method)

Purpose Computation of dissipation coefficientand the nondimensional group velocity ofharmonic wave (11) for various schemes.

Input parameters kh Product of wave number by space stepr Courant numbermethod Method

'crs' scheme cross'lxw' Lax-Wendroff scheme'roe' scheme of P. Roe

'god' Godunov scheme'pem' PEM scheme

Output parameters dis Dissipation coefficientcg Nondimensional group velocity

function [uu]=cross(ud,um,c,ro,nx,h,time,tau,bl,br,fbl,fbr)

Purpose Computation of one time step from

scheme (6)

Input parameters ud, um Solution at the instants of times tk-1tk,respectivel.

c Speed of sound

ro Density

nx Number of grid nodesh Space steptime Instant of time tk+1tau Time stepbl,br Type of boundary condition at x=0x=l,

respectively.=0 boundary condition of the 1st kind=1 boundary condition of the 2ndkind.

fbl, fbr Names of m-files in which thedependence of a solution or its derivative


Output parameters uu Solution at the instant of time tk+1


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function [uu,pu]=lw_acoustics(ud,pd,imp,nx,time,r,bl,br,fbl,fbr)

Purpose Computation of one time step from Lax-Wendroff scheme (7)

Input parameters ud, pd Velocity and pressure at the instant oftime tk

imp Value ofcnx Number of grid nodestime Instant of time tk+1r Courant numberbl,br Type of boundary condition at x=0x=l,

respectively.=0 boundary condition on u=1 boundary condition on p

fbl, fbr Names of m-files in which thedependence of velocity or pressure on

time is defined at x=0 and x=l,respectively.

Output parameters uu, pu Velocity and pressure at the instant oftime tk+1

function [uu,pu]=roe(ud,pd,um,pm,imp,nx,time,r,bl,br,fbl,fbr)

Purpose Computation of one time step from

scheme of P. Roe (8)

Input parameters ud, pd Velocity and pressure at the instant oftime tk-1

um, pm Velocity and pressure at the instant oftime tk

imp Value ofc

nx Number of grid nodestime Instant of time tk+1r Courant numberbl,br Type of boundary condition at x=0x=l,






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function [uu,pu]=godunov_acoustics(ud,pd,imp,nx,time,r,bl,br,fbl,fbr)

Purpose Computation of one time step fromGodunov scheme (9)

Input parameters ud, pd Velocity and pressure at the instant oftime tk

imp Value ofcnx Number of grid nodestime Instant of time tk+1r Courant numberbl,br Type of boundary condition at x=0x=l,






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function [ub,pb,uu,pu]=pem(ua,pa,ud,pd,imp,nx,r,time,tau,a1,a2,a3,b1,b2,b3,bl,br,fbl,fbr)

Purpose Computation of one time step from PEMscheme (10)

Input parameters ua, pa Average values of velocity and pressureon every difference interval at the instantof time tk

ud, pd Values of velocity and pressure in everynode of a grid at the instant of time tk

imp Value ofcnx Number of grid nodestime Instant of time tk+1R Courant numbertau Time stepa1,a2,a3,b1,b2,b3

Parameters of PEM scheme (given in

example5_5.m)

bl,br Type of boundary condition at x=0x=l,respectively.

=0 boundary condition on u=1 boundary condition on p

fbl, fbr Names of m-files in which thedependence of velocity or pressure ontime is defined at x=0 and x=l,respectively.

Output parameters ub, pb Average values of velocity and pressureon every difference interval at the instant

of time tk+1 uu, pu Values of velocity and pressure in every

node of a grid at the instant of time tk+1


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22

function [rf,uf,pf]=riemann(x,t,r1,u1,p1,r2,u2,p2,gamma)

Purpose Computation of the exact solution ofRiemann problem for an ideal gas

Input parameters x Coordinatet Timer1,u1,p1 Initial density, velocity and pressure at

x0gamma Adiabatic exponent

Output parameters rf,uf,pf Density, velocity and pressure at x and atinstant of time t

function [d,u,p,e,time]=lw_gasdynamics(dd,ud,pd,h,nx,r,cad,tp,gamma,bcl,bcr)

Purpose Computation of a gas flow from Lax-

Wendroff scheme (12)

Input parameters dd,ud,pd Density, velocity and pressure at time t=0h Space stepnx Number of grid nodesr Courant numbercad Artificial viscosity coefficienttp Flow duration (approximate)gamma Adiabatic exponentbcl,bcr Type of boundary condition at x=0x=l,

respectively.=0 rigid wall=1 open boundary

Otput parameters time Real duration of the flowd,u,p,e Density, velocity, pressure and specific

internal energy at time time


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function [d,u,p,e,time]=godunov_gasdynamics(dd,ud,pd,h,nx,r,tp,gamma,bcl,bcr)

Purpose Computation of a gas flow from Godunovscheme (13)

Input parameters dd,ud,pd Density, velocity and pressure at time t=0h Space stepnx Number of grid nodesr Courant numbertp Flow duration (approximate)gamma Adiabatic exponentbcl,bcr Type of boundary condition at x=0x=l,




function [d,u,p,e,time]=waf_gasdynamics(dd,ud,pd,h,nx,r,cad,tp,gamma,bcl,bcr)

Purpose Computation of a gas flow from the WAFscheme (15).

Input and output parameters have the same meaning as for the function lw_gasdynamics.


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function [d,u,p,e,time]=waf_tvd_gasdynamics(dd,ud,pd,h,nx,r,tp,gamma,bcl,bcr,limiter)

Purpose Computation of a gas flow from a TVDversion of the WAF scheme

Input parameters dd,ud,pd Density, velocity and pressure at time t=0h Space stepnx Number of grid nodesr Courant numbertp Flow duration (approximate)gamma Adiabatic exponentbcl,bcr Type of boundary condition at x=0x=l,


limiter Name of m-file in which a limiterfunction is defined. Several limiters are

enclosed.




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function [u,k]=helmgoltz_2d_sor(f,ac,bc,lx,ly,nx,ny,g1,g2,g3,g4)

Purpose Computation of a solution of system (18)with Dirichlet boundary conditions,method SOR.

Input parameters f Name of m-file in which the right-hand

side function f(x, y) is defined.ac Name of m-file in which the coefficient

a(x, y) is defined.bc Name of m-file in which the coefficient

b(x, y)is defined.

lx, ly Domain size in x y directions,respectively.

nx, ny Number of grid nodes in x ydirections, respectively.

g1, g3 Names of m-files in which thedependence of a solution on y is definedat x=0 andx=lx, respectively.

g2, g4 Names of m-files in which thedependence of a solution on x is definedat y=0 andy=ly, respectively.

Output parameters u Approximate solutionk Number of iterations

Note Iterative process is completed when

either relative error of approximate

solution becomes less then 10-5

, ornumber of iterations reaches the value

3max(nx, ny).


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function [u]=helmgoltz_2d_fft(f,a,b,lx,ly,nx,ny,g1,g2,g3,g4)

Purpose Computation of a solution of system (16)with Dirichlet boundary conditions, FFT-based method.

Input parameters f Name of m-file in which the right-hand

side function f(x, y) is defined.a Name of m-file in which the coefficient

a(x, y) is defined.B Name of m-file in which the coefficient

b(x, y)is defined.





Output parameters u Approximate solution

Note FFT is applied in x direction, therefore,one should set nx=2p+1 for the mostefficient computation.


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function [u,it]=helmgoltz_2d_cg(up,f,a,b,lx,ly,nx,ny,g1,g2,g3,g4)

Purpose Computation of a solution of equation(17) with Dirichlet boundary conditions.Method of Concus and Golub (19) based

on FFT.

Input parameters up Initial approximationf Name of m-file in which the right-hand


a(x, y) is defined.b Name of m-file in which the coefficient

b(x, y) is defined.lx, ly Domain size in x y directions,

respectively.




Output parameters u Approximate solutionit Number of iterations

Note Iterative process is completed whenrelative error of approximate solution

becomes less then 10-5

.FFT is applied in x direction, therefore,one should set nx=2

p+1 for the mostefficient computation.


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function [u,k]=poisson_2d_td(up,f,a,lx,ly,nx,ny,g1,g2,g3,g4)

Purpose Computation of a solution of nonlinearequation (20) with Dirichlet boundary

conditions, method of time development(21)

Input parameters up Initial approximationf Name of m-file in which the right-hand


a(u, x, y) is defined.lx, ly Domain size in x y directions,

respectively.




Output parameters u Approximate solutionk Number of iterations

Note Iterative process is completed whenrelative error of approximate solution

becomes less then 10-5

.


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function [u,v,w,psi,time]=ns_2d_vsf(u,v,w,Re,tp,lx,ly,nx,ny,bxl,bxr,byl,byr,g1,g2,g3,g4)

Purpose Computation of a solution of equations(22) from the scheme (23)-(25).

Intput parameters u, v,w Components of velocity and vorticity attime t=0.

Re Reynolds number

tp Flow duration (approximate)



bxl,bxr Type of boundary condition on v at x=0andx=lx, respectively:=0 value of velocity is assigned=1 v/y is zero.

byl,byr Type of boundary condition on u at y=0andy=ly, respectively:=0 value of velocity is assigned,=1 u/x is zero.

g1, g3 Names of m-files in which thedependence ofv on y and time is defined

at x=0 andx=lx, respectively.g2, g4 Names of m-files in which thedependence ofu on x and time is definedat y=0y=ly, respectively.

Output parameters time Real duration of the flowu,v,w Components of velocity and vorticity at

time time.psi Stream function at time time.

Note For normal to the boundaries componentsof velocity the following conditions are

assumed:u=0 at x=0 andx=lx, v=0 at y=0 andy=ly


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References

1. Concus P. and Golub G. H. Use of fast direct methods for the efficient numerical solution of non-separableelliptic equations, SIAM J. Num. Anal., v. 10, pp. 1103-1120 (1973).

2. Roe P. Linear bicharacteristic scheme without dissipation, SIAM J. Sci. Comput., v.19, N5, pp. 1405-1427(1998).

3. Toro E. F. Riemann solvers and numerical methods for fluid dynamics, Springer-Verlag, 1997.

4. Zalizniak V. The piecewise exponential method (PEM) for the numerical simulation of wave propogation,Communications in Numerical Methods in Engineering, v. 13, N3, pp163-171.

Documents

ECP1 Software Manual