mandeep singh cfd

8/7/2019 Mandeep Singh cfd

1/19

[Project#04:1D Wave

Equation]

[MAE]

[542]

[EngineeringApplicationsofComputationalFluidDynamics][ 19th April

2011]

Submittedby

MandeepSingh

Person#37212672


2/19

2|P a g e

Contents1 Introduction.................................................................................................................................... 3

2 Problemdefinition/Problemstatement....................................................................................... 3

3 MethodofSolution:....................................................................................................................... 5

4 Discussionofresults....................................................................................................................... 6

4.1 Case1.Usingthevalueofc=0.6,x=0.025,t=1............................................................... 6

4.2 Case2.Usingthevalueofc=0.3,x=0.025,t=1............................................................... 8

4.3 Case3.Usingthevalueofc=0.6,x=0.01,t=1................................................................. 9

5 ConclusionandDiscussion............................................................................................................ 11

6 Appendix....................................................................................................................................... 13

6.1 MatlabCode.......................................................................................................................... 13

7 References.................................................................................................................................... 19


3/19

3|P a g e

1 Introduction

Waveequation: It isabasemodelfortheHyperbolicequations.Hyperbolicpartial

differentialequationshavetimederivativeintheequation.Someofthemethodsto

solve this equation in time marching are Lax Wendroff, Upwind , Lax Wendroff

method with flux limiter method using control volume like smooth limiter and

superbee limiter andmidpoint leapfrogmethodsaresomeofexplicitmethods

forexploringthe wavepropagation.

The

1D

linear

wave

equation

is

given

by

equation

0x

u

t

u=

+

a

(1)

Whereuisthescalarfunctionsatisfyingthewaveequationandxisthespace

domain.aisthespeedofpropagation.

2

Problemdefinition

/

Problem

statement

In this project we need to solve the given wave equation for periodic boundary

condition.ThedomainlengthL=1.

PeriodicBoundaryconditionsforthegivenproblemaregivenasfollows:

Lxx

xLx

txutxu

txutxu

==

==

=

+=+

|),(|),(

|),(|),(

0

0

Thevalueofa=1.Weneedtoanalyzetwoinitialcondition,Heavisidesharpstep

function,whereinitialconditionisgivenasuo(x)=1H(x 0.5),whereHisthe

Heavisidefunction.


4/19

4|P a g e

Weneed todevelop a computeralgorithm to solve the givenhyperbolicequation

using

the

Lax

Lax

Wendroff

method

without

flux

limiter,

Upwind

Method

,

Lax

wendroffwithFluxlimiter(superbeeandsmooth).

Weneedtoanalyzethesystemusingthefollowingvariationinthesystem

S. No Condition to Explore Initial Condition Parameter given

1 ComparingLaxWendroffwithoutlimiter,

Upwindmethod,LaxWendroffwithsmoothlimiter,LaxWendroffwith

superbeelimiter,analytical

uo(x)=1H(x 0.5) c=0.6, t=1.0,

x=0.025


Upwindmethod,LaxWendroffwith

smoothlimiter,LaxWendroffwith

superbeelimiter,analytical

uo(x)=1H(x 0.5) c=0.3, t=1.0,

x=0.025


Upwindmethod,LaxWendroffwith

smoothlimiter,LaxWendroffwithsuperbeelimiter,analytical

uo(x)=1H(x 0.5) c=0.6, t=1.0,

x=0.01


5/19

5|P a g e

3 MethodofSolution:

In thisprojectwearegiven the followingalgorithms for theLaxWendroffmethod

andupwindmethodforwhichthefluxlimitervalueareassignedtocalculatethewe

willusethetwoalgorithmsgiveni.e.,LaxandLaxWendroffforgettingthenumerical

solutionforthegivenequation.LaxAlgorithmisgivenbytheequation:

x

tFFuu

n

i

n

i

n

i

n

i

= ++

+

2

1

2

11

2

1

1

2

12

)1(+

+

+

=

i

n

i

n

in

i

n

i

uucaauF

2

1

1

1

2

12

)1(

=

i

n

i

n

in

i

n

i

uucaauF

n

i

n

i

n

i

n

in

i uu

uu

=

+

+1

1

2

1

n

i

n

i

n

i

n

in

i uu

uu

1

21

2

1

=

Wherecisthecourantnumbergivenbytheequationc=a t/x, isthefluxlimiter

whichisthefunctionof.


6/19

6|P a g e

ForLaxWendroffmethodwithoutlimiter, =1,forupwindmethod =0,forLax

Wendroffwith smooth limiter()= (||+ )/(1+||) ,and forLaxWendroffwith

smoothlimiter()=max(0,min(1,2),min(,2))

4 Discussionofresults

4.1 Case1.Usingthevalueofc=0.6,x=0.025,t=1For all themethodsusing thedefinedparameters in case 1,we get the following

solutioncomparingallthemethodsasshowninthefigure1.

Figure1(Figureshowingthesolutionforthewaveequationforthevariousmethodsatc=0.6,Deltax=0.025)


7/19

7|P a g e

From the figure 1, we can observe that the solution for the wave propagation

function is mapped best through the superbee function with least diffusion error

when compared to the other methods. The smooth limiter follows the superbee

method, with the second best solution for the wave equation with the give

parameters.WecanstillseealargeamountofdispersionerrorintheLaxWendroff

withoutlimiterwhichcreateanerrorof20%atcertainnodalsolutions.Thisisdueto

theodd spatialderivative in the truncation error that causesdispersion. Thishigh

orderconvectiontendstodistorttheshapeofthepropertydistributioninspaceand

causeswigglesinthesolution.Fortheupwindscheme,which ishighlydissipativein

nature,gives the leastaccurateapproximation to thewaveequation.For instance,

suppose at length L = 0.1,upwind scheme gives 18% error in the solution.At the

sameinstanceLaxWendroffwithoutlimitergivesanderrorof6%.TheLaxWendroff

withsmooth limitergivesanerrorofaround3%andtheLaxWendroffgivesalmost

negligibleerroratthegivenlocation.


8/19

8|P a g e

4.2 Case2.Usingthevalueofc=0.3,x=0.025,t=1For the value of c = 0.3, we get the following solution as shown in the figure 2.


Forthegivenconditionoftheparameters,wefindthattheinaccuracyoftheallthe

methods increases. Still the solutions from Lax Wendroff with smooth limiter and

superbee limiterarecomparable.TheLaxWendroffwithno flux limitersuffers the

dispersionerror in the solution. For the LaxWendroffwithno flux limiterwe can

observe thaat location0.35, thediffusionerror increases to22%which is greater

thantheformercasewherethediffusionerrorwas20%.Thisisduetothefacethat

thevalueofchasbeendecreasedto0.3from0.6.AtDomainlengthL=0.02,wecan

observe a wiggle growing up for the Lax Wendroff method without limiter. The

diffusionerrorintheupwindschemehasincreasedby4%fromtheearliercase.Still

theLaxWendroffwiththesuperbeelimitergivesthebestresultswhencomparedto


9/19

9|P a g e

other followed by smooth limiter method. Both Lax Wendroff superbee and Lax

Wendroffsmooth limitergivesmorestablesolutionwith leastdiffusionerror inthe

solution.Alsonoshiftingwasobservedforthesuperbee.

4.3 Case3.Usingthevalueofc=0.6,x=0.01,t=1

Incase3weraisedthevalueofcagainto0.6comparedto0.3incase2i.e.,equalto

case 1. But at the same time we have decreases the space stepx to 0.01. By

decreasing the valueofx from 0.025 to 0.01 we can clearly see a considerable

enhancementinthesolutionasshowninFig.3.


10/19

10|P a g e


Incase3,weasshowninfigure3.WecansseestilltheLaxWendroffmethodwith

superbee gives the best approximation followed by the smooth limiter. The Lax

Wendroffwithout limiterstillowes thedispersiveerror in thesolutionwithalmost

sameamountofashighas20%atsomenodes.Theupwindschemeerrorhasalso

reducedandapproximatestheprofileofthesolution.Hencebydecreasingthevalue

ofx from0.025to0.01,solutionsforallmethodsimprovedexceptforthe caseof

LaxWendroffwithnolimiter,wherewestillgetthewigglesofsameintensityasfor

thecase1and2.


11/19

11|P a g e

5 ConclusionandDiscussionInthisprojectwestudiedthebehavioroftheLaxWendroff fluxlimiterschemes.We

observedthattheaccuracyforthelaxwendroffmethodusingthesuperbeemethod

isquitestablewithrespecttodiffusionerrorandconsiderablyaccurate.Itisfollowed

bytheLaxWendroffmethodwithsmoothlimiter,whichwascomparativelyaccurate

thantheupwindandtheLaxWendroffwithnolimiter.FluxLimitersareusedinhigh

resolutionschemeswhichavoidthewigglesinthesolution.Thesewigglesoccurdue

tosharpchanges inthewavepropagation ,anydiscontinuitiesorshock.Hencewe

haveobservednowigglesfortheLaxWendroffmethodwithsuperbeeandsmooth

limiterandatthesametimemaintainingconsiderableaccuracyevenatlowvalueof

c(courantnumber).Theupwind schemewas found tobequitedissipative.TheLax

Wendroff method have quite sensitive behavior for the abrupt changes in the

boundary.Forthisreasonweobservedhighdispersiveerrorduringthesolutionfor

theHeavisideboundarycondition.Themainreasonforthisdifferent behaviorinthe

two methods is because of the dispersive leading truncation error in the Lax

Wendroffmethod.Flux limiters (slope limiters) ,asobserved,avoided thespurious

oscillations (wiggles) that were otherwise occurring in Lax Wendroff scheme with

high order spatial discretisation schemes due to shocks, discontinuities or sharp

changes in the solution domain. Hence use of flux limiters, together with an

appropriateschememakesthesolutiontotalvariationdiminishing(TVD).Wecansay

that the superbee scheme was the best among the all other scheme for this

heavyside step functions wave. But at the same time , it is known to suffer

sharpening of the slopes when approximating the smooth curves like sinusoidal

waves.


12/19

12|P a g e

TheswebyTVDplotshowsthefluxlimiterareaforthevariousmethods(figure4).Itis

plottedusingthevaluesofvaryingthevaluesof forvariousmethods.

Figure4(swebyplotforcomparisonofTVDandnonTVDschemes)

The Superbee limiter applies the minimum limiting and maximum steepening

possible to remain TVD. It is known to suffer from excessive sharpening of

slopesasa result. ThevanLeer(Laxwendroffwithsmooth limiter) limiterchartsa

careful compromise path through the Sweby TVD region. Lax Wendroff without

limiter runs forconstantwith thevalueof=1.HenceLinearnonTVD limiters like

upwind , Lax Wendroff have a constant same slope. The superbee and the

smooth(vanleer)nonlinearsecondorderTVDlimiterhavevaryingslopes.


13/19

13|P a g e

6 Appendix6.1 MatlabCode

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Name : Mandeep Singh %% Project 4 (1-D Wave Equation using Non linear Methods) %% Assigned: 04/05/11 Due: 04/19/11 %% MAE 542 Engineering Applications of Computational Fluid Dynamics %% Flux Limited Lax Wendroff method %% for solving 1D Wave Equation contrl volume method %% Associated file : Myinput.m %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%------------------------------------------------------------------%

% given% Equation du/dt + a (du/dx) = 0% Wave velocity a = 1% given the initial boundary condition% and uo= 1 - H(x-0.5)

%%clear allclose allclc%% Inputs:t=myinput('Please Enter maximum time for iternation(t=1):',1);dt=myinput('Please Enter time step dt(dt=0.015):',0.015);

dx=myinput('Please Enter the value of space step dx(dx=0.025):',0.025);L=myinput('Please Enter the lenght (L=1):',1);a=myinput('Please Enter the value of waves speed a(a=1):',1);

nt=((t/dt)+1);%Time steps number

nx= (L/dx)+1;%Space steps number

x=0:dx:L;

for i =1:length(x)uo(i)= 1-heaviside(x(i)-0.5);end%initial boundary conditionuo(:,1)=0.5;uo(:,end)=0.5;u=zeros(nx,nt);%Initializing the

% lax method

c=a*dt/dx;


14/19

14|P a g e

% Lax Wendroff Method without limiterUA=zeros(nx,nt);UB=zeros(nx,nt);

UC=zeros(nx,nt);UD=zeros(nx,nt);UA(:,1)=uo;UB(:,1)=uo;UC(:,1)=uo;UD(:,1)=uo;

%% ------------------------------------------------------------------------%% Case A ; phi = 1%% ----------------

for i=1:nt %0:length(t)for j=1:nx %0:length(x)

if j==1||j==nxphi1=1;phi2=1;Fplus(j,i)=a*UA(j,i)+((a*(1-c)/2)*(UA(2,i)-UA(j,i)))*phi1;Fminus(j,i)=a*UA(nx-1,i)+(a*(1-c)/2)*(UA(j,i)-UA(nx-1,i))*phi2;UA(j,i+1)=UA(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

elsephi1=1;

phi2=1;Fplus(j,i)=a*UA(j,i)+((a*(1-c)/2)*(UA(j+1,i)-UA(j,i)))*phi1;Fminus(j,i)=a*UA(j-1,i)+(a*(1-c)/2)*(UA(j,i)-UA(j-1,i))*phi2;UA(j,i+1)=UA(j,i)-(Fplus(j,i)*phi1-Fminus(j,i)*phi2)*(dt/dx);

endend

end

%% ------------------------------------------------------------------------%% ------------------------------------------------------------------------%% Case B ; phi = 0%% ----------------for i=1:nt %0:length(t)

for j=1:nx %0:length(x)if j==1||j==nx

phi1=0;phi2=0;Fplus(j,i)=a*UB(j,i)+((a*(1-c)/2)*(UB(2,i)-UB(j,i)))*phi1;Fminus(j,i)=a*UB(nx-1,i)+(a*(1-c)/2)*(UB(j,i)-UB(nx-1,i))*phi2;UB(j,i+1)=UB(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

elsephi1=0;

phi2=0;


15/19

15|P a g e

Fplus(j,i)=a*UB(j,i)+((a*(1-c)/2)*(UB(j+1,i)-UB(j,i)))*phi1;Fminus(j,i)=a*UB(j-1,i)+(a*(1-c)/2)*(UB(j,i)-UB(j-1,i))*phi2;UB(j,i+1)=UB(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

endend

end

%% ------------------------------------------------------------------------%% ------------------------------------------------------------------------%% Case C%% ------

for i=1:nt %0:length(t)for j=1:nx %0:length(x)if j==1||j==nx

theta1(j,i)=(UC(j,i)-UC(nx-1,i))/((UC(2,i)-UC(j,i))+10^-5);theta2(j,i)=(UC(nx-1,i)-UC(nx-2,i))/((UC(j,i)-UC(nx-1,i))+10^-5);

phi1(j,i)=((abs(theta1(j,i))+theta1(j,i))/(1+abs(theta1(j,i))));phi2(j,i)=((abs(theta2(j,i))+theta2(j,i))/(1+abs(theta2(j,i))));

Fplus(j,i)=a*UC(j,i)+((a*(1-c)/2)*(UC(2,i)-UC(j,i)))*phi1(j,i);Fminus(j,i)=a*UC(nx-1,i)+(a*(1-c)/2)*(UC(j,i)-UC(nx-1,i))*phi2(j,i);UC(j,i+1)=UC(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

elseif j==2

theta1(j,i)=(UC(j,i)-UC(j-1,i))/((UC(j+1,i)-UC(j,i))+10^-5);theta2(j,i)=(UC(j-1,i)-UC(nx,i))/((UC(j,i)-UC(j-1,i))+10^-5);

phi1(j,i)=((abs(theta1(j,i))+theta1(j,i))/(1+abs(theta1(j,i))));phi2(j,i)=((abs(theta2(j,i))+theta2(j,i))/(1+abs(theta2(j,i))));

Fplus(j,i)=a*UC(j,i)+((a*(1-c)/2)*(UC(j+1,i)-UC(j,i)))*phi1(j,i);Fminus(j,i)=a*UC(j-1,i)+(a*(1-c)/2)*(UC(j,i)-UC(j-1,i))*phi2(j,i);UC(j,i+1)=UC(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

else

theta1(j,i)=(UC(j,i)-UC(j-1,i))/((UC(j+1,i)-UC(j,i))+10^-5);theta2(j,i)=(UC(j-1,i)-UC(j-2,i))/((UC(j,i)-UC(j-1,i))+10^-5);

phi1(j,i)=((abs(theta1(j,i))+theta1(j,i))/(1+abs(theta1(j,i))));

phi2(j,i)=((abs(theta2(j,i))+theta2(j,i))/(1+abs(theta2(j,i))));

Fplus(j,i)=a*UC(j,i)+((a*(1-c)/2)*(UC(j+1,i)-UC(j,i)))*phi1(j,i);Fminus(j,i)=a*UC(j-1,i)+(a*(1-c)/2)*(UC(j,i)-UC(j-1,i))*phi2(j,i);


16/19

16|P a g e

UC(j,i+1)=UC(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

endend

end

%% ------------------------------------------------------------------------%% ------------------------------------------------------------------------%% Case D

for i=1:nt %0:length(t)for j=1:nx %0:length(x)if j==1||j==nx

theta1(j,i)=(UD(j,i)-UD(nx-1,i))/((UD(2,i)-UD(j,i))+10^-10);theta2(j,i)=(UD(nx-1,i)-UD(nx-2,i))/((UD(j,i)-UD(nx-1,i))+10^-10);

jp1=max(min(theta1(j,i),2),min(1,2*theta1(j,i)));phi1=max(0, jp1);


Fplus(j,i)=a*UD(j,i)+((a*(1-c)/2)*(UD(2,i)-UD(j,i)))*phi1;Fminus(j,i)=a*UD(nx-1,i)+(a*(1-c)/2)*(UD(j,i)-UD(nx-1,i))*phi2;UD(j,i+1)=UD(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

elseif j==2

theta1(j,i)=(UD(j,i)-UD(j-1,i))/((UD(j+1,i)-UD(j,i))+10^-10);theta2(j,i)=(UD(j-1,i)-UD(nx,i))/((UD(j,i)-UD(j-1,i))+10^-10);



Fplus(j,i)=a*UD(j,i)+((a*(1-c)/2)*(UD(j+1,i)-UD(j,i)))*phi1;Fminus(j,i)=a*UD(j-1,i)+(a*(1-c)/2)*(UD(j,i)-UD(j-1,i))*phi2;UD(j,i+1)=UD(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

else

theta1(j,i)=(UD(j,i)-UD(j-1,i))/((UD(j+1,i)-UD(j,i))+10^-10);theta2(j,i)=(UD(j-1,i)-UD(j-2,i))/((UD(j,i)-UD(j-1,i))+10^-10);

jp1=max(min(theta1(j,i),2),min(1,2*theta1(j,i)));


17/19

17|P a g e

phi1=max(0, jp1);


Fplus(j,i)=a*UD(j,i)+((a*(1-c)/2)*(UD(j+1,i)-UD(j,i)))*phi1;Fminus(j,i)=a*UD(j-1,i)+(a*(1-c)/2)*(UD(j,i)-UD(j-1,i))*phi2;UD(j,i+1)=UD(j,i)-(Fplus(j,i)-Fminus(j,i))*(dt/dx);

endend

end

%% ------%% ------------------------------------------------------------------------%% ------------------------------------------------------------------------

for i=1:nxif i==1||i==nx||i==(nx+1)/2

uanal(i) = 0.5elseif i > 1 && i


18/19


19/19

19|P a g e

7 References1. Lecture

notes

by

Prof.

Desjardin

2. Hoffman&Chian,ComputationalFluidDynamicsVolI

3. NumericalMethods

mandeep singh cfd

Documents