Investigated the topic Discretisation Schemes, most notably, Finite Difference Approximations and there ability to accurately provide solutions for Partial Differentials. Three methods in-particular were executed, Forward-Time Backward-Space, Lax Scheme and the Lax-Wendroff Scheme varying key parameters to establish operation supremacy.

1. Solution of Partial Differential Equations by Finite-Difference
Approximations
Aeronautical Engineering (MEng)
Thursday, 23 July 2015
Elliot Newman
@00320195
Word Count: 7916

2. P a g e | i
Contents
Contents.......................................................................................................................................i
Tables..........................................................................................................................................ii
Figures.........................................................................................................................................ii
Introduction.................................................................................................................................1
Objectives ................................................................................................................................ 1
Theory.........................................................................................................................................3
Partial Differentials................................................................................................................... 3
Discretisation Schemes ............................................................................................................. 4
Finite-Difference Approximation................................................................................................ 4
Forward-Time, Backward-Space Scheme................................................................................. 5
Lax Scheme........................................................................................................................... 5
Lax-Wendroff Scheme ...........................................................................................................5
Taylor Series............................................................................................................................. 6
Procedure....................................................................................................................................7
Forward-Time, Backward-Space Scheme.................................................................................... 7
Lax Scheme .............................................................................................................................. 7
Lax-Wendroff Scheme............................................................................................................... 8
Results.........................................................................................................................................9
Forward-Time, Backward Space Scheme .................................................................................... 9
Lax Scheme ............................................................................................................................ 12
Lax-Wendroff Scheme............................................................................................................. 15
Analysis...................................................................................................................................... 19
Discussion.................................................................................................................................. 21
Forward-Time, Backward-Space Taylor Series Expansion........................................................... 21
FT-BS Truncation Error......................................................................................................... 21
Lax Scheme Taylor Series Expansion......................................................................................... 21
Lax Truncation Error............................................................................................................ 22
Lax-Wendroff Scheme Taylor Series Expansion......................................................................... 22
Lax-Wendroff Truncation Error............................................................................................. 22
Conclusion ................................................................................................................................. 26
References................................................................................................................................. 27

3. P a g e | ii
Tables
Table 1: Forward-Time, Backward-Space computational grid (𝒗 = 𝟎. 𝟓)...........................................9
Table 2: Forward-Time, Backward-Space Scheme computational grid. (𝒗 = 𝟎. 𝟓𝟏)......................... 11
Table 3: Forward-Time, Backward-Space Scheme computational grid. (𝒗 = 𝟎)............................... 11
Table 4: Forward-Time, Backward-Space Scheme computational grid. (𝒗 = −𝟎. 𝟎𝟏) ...................... 11
Table 5: Lax Scheme computational grid (𝒗 = 𝟎. 𝟓)....................................................................... 12
Table 6: Lax Scheme computational grid (𝒗 = 𝟎. 𝟓𝟏).................................................................... 13
Table 7: Lax Scheme computational grid (𝒗 = −𝟎. 𝟓).................................................................... 14
Table 8: Lax Scheme computational grid (𝒗 = −𝟎. 𝟓𝟏).................................................................. 14
Table 9: Lax Scheme computational grid (𝒗 = 𝟎).......................................................................... 15
Table 10: Lax-Wendroff Scheme computational grid (𝒗 = 𝟎. 𝟓)..................................................... 15
Table 11: Lax-Wendroff Scheme computational grid (𝒗 = 𝟎. 𝟓𝟏)................................................... 16
Table 12: Lax-Wendroff Scheme computational grid (𝒗 = −𝟎. 𝟓𝟎)................................................ 17
Table 13: Lax-Wendroff Scheme computational grid (𝒗 = −𝟎. 𝟓𝟏)................................................ 17
Table 14: Lax-Wendroff Scheme computational grid (𝒗 = 𝟎)......................................................... 18
Table 15: Time stepvariation FT-BS. ............................................................................................ 23
Figures
Figure 1: Initial Conditions temperature step graph........................................................................2
Figure 2: Transport example of PDE function (University of Alaska, n.d.)..........................................3
Figure 3: Forward-Time, Backward-Space Scheme step graphs. (𝒗 = 𝟎. 𝟓) ..................................... 10
Figure 4: Forward-Time, Backward-Space Scheme step graphs. (𝒗 = 𝟎. 𝟓𝟏)................................... 10
Figure 5: Forward-Time, Backward-Space Scheme step graphs. (𝒗 = 𝟎)......................................... 11
Figure 6: Forward-Time, Backward-Space Scheme step graphs. (𝒗 = −𝟎. 𝟎𝟏) ................................ 11
Figure 7: Lax Scheme step graphs(𝒗 = 𝟎. 𝟓)................................................................................. 13
Figure 8: Lax Scheme step graphs(𝒗 = 𝟎. 𝟓𝟏).............................................................................. 13
Figure 9: Lax Scheme step graphs(𝒗 = −𝟎. 𝟓).............................................................................. 14
Figure 10: Lax Scheme step graphs (𝒗 = −𝟎. 𝟓𝟏).......................................................................... 14
Figure 11: Lax Scheme step graphs (𝒗 = 𝟎).................................................................................. 14
Figure 12: Lax-Wendroff Scheme step graphs (𝒗 = 𝟎. 𝟓)............................................................... 16
Figure 13: Lax-Wendroff Scheme step graphs (𝒗 = 𝟎. 𝟓𝟏)............................................................. 16
Figure 14: Lax-Wendroff Scheme step graphs (𝒗 = −𝟎. 𝟓𝟎).......................................................... 17
Figure 15: Lax-Wendroff Scheme step graphs (𝒗 = −𝟎. 𝟓𝟏).......................................................... 17
Figure 16: Lax-Wendroff Scheme step graphs (𝒗 = 𝟎)................................................................... 17
Figure 17: Percentage error of Lax Method (𝒗 = 𝟎)...................................................................... 24
Figure 18: Relative overshoot error percentage (𝒗 = 𝟎. 𝟓𝟏).......................................................... 24

4. P a g e | 1
Introduction
Aerodynamics,beingamulti-facetedsubject, manifestsitself inmanyforms, mostnotably in
understandingthe complex motionsthe airtranscendsthroughwhilstunder the influence of an
applieddevice.Yielded knowledge canthenbe impartedintothe designprocessestostimulate the
nextgenerationof more efficientaerodynamicdevices.Thiscanbe achievedthroughamultitude of
techniques,althoughthispapershall focuson strictlynumerical approaches,solvingthe differentials
throughfinite-difference approximations. These methodsallow forcomplex differentialsthatare
eitherunsolvable orverydifficulttodoso,to be alludedtonumerically,withthesesystems
providingsuitable levelsof accuracythat these resultscanthenbe carriedforwardto further
calculations of the aerodynamicbehavioursunderinvestigation,whetherthatbe, flow velocity,drag
levelsorinthiscase linearconvection.
Objectives
Overthe course of thisassignment,finite difference approximationsshall be investigatedandtheir
relevanttheorycollatedbeforefocusingonthree methodsinparticular:
 ForwardTime,BackwardSpace Scheme,a backward systemderiving thenextdata setof the
previously neighbouring geometricnodes(page5).
 Lax Scheme,a centralised method,using pointsaboveand below in the previoustime step
(page5).
 Lax Wendroff Scheme,a symmetricpracticecombining theprevioustwo and thereforeusing
three points to calculatethe nextstep (page5).
These three approximationmethodsshall thenbe testedand implemented ontothe computational
domainforpartial differential equationforlinearconvection:
𝜕𝑇
𝜕𝑡
+ 𝑈
𝜕𝑇
𝜕𝑥
= 0
Where 𝑇 is the Temperaturechangingwithtime, 𝑡 and 𝑈 equatestothe ConstantConvection
Velocityin the 𝑥-direction, varyingwith 𝑥.
The final results canthenbe comparedto allude towhich systemisthe mostaccurate and effective.
Achievingthisdirectiverequiresfurtherinvestigations intotheirconsistencyandstability,whichin
turn have an effectonconvergence, throughthe variance of akeyparameter, 𝑣 =
𝑈∆𝑡
2∆𝑥
,with
∆𝑡
∆𝑥
correspondingtothe stabilitycoefficient,wherein-betweenthe boundaryvaluesthe predictionis
stable.The instabilityof asystembecomesnoticeablewhenthe extrapolatingvaluesoscillate
violentlyandextrude awayfromthe previousstep.Movingthroughthe time stepsisalsoonthe
agendaand anotherareaof comparison,viewingthe equationsastheytranscendsovertime
periods. Throughthisprocessof testing,eachschemesadvantagesanddisadvantagescanalsobe
alludedtoandhelpto differentiate betweenwhetherasystemissuperiortoanotheror simply
bettersuitedtothe situation,valuable andapplicable informationmovingforwards.
A setof initial conditionshave beenprovidedandpredeterminedinordertocarry out the
investigation:
At time 𝑡 = 0,the temperature atthe 21 finite-difference gridpointsare asfollows:
𝑇1 = 0.0, 𝑇2 = 1.0, 𝑇3 = 2.0, 𝑇4 = 2.0, 𝑇5 = 1.0

5. P a g e | 2
𝑇𝑖 = 0.0, for 𝑖 = 6 𝑡𝑜 21.
Withthe final boundaryconditionsthatpoints, 𝑇1 and 𝑇21 (bothendpoints) remainfixedat 0.0 for
all time, 𝑡.
An appropriate setof initial andboundaryconditionsare vital tothe user’sabilitytosolve anypartial
differential,asif the problemisoverconstrainedthenthere willbe nosolutions,yetleftsuitably
undefined,thenthere willnotbe anyunique solutionstothe problem.
Withthe appropriate conditionssetandthroughcollatingandimplementingthe initial conditions
the followingstartingstep graph(n) isgenerated:
Thisgraph representsthe solutiontothe initial conditionsandaperfectlyaccurate approximation
methodwouldcontinue topredictthissteppatternthroughthe entiretyof the perceivedwindow.
As well astestingthe accuracyof eachmethod,throughuse of the Taylor Series,anumerical
expansion practice toapproximatepartial differentialswhichisdiscussedlater,theirconsistencycan
be highlightedasthroughthe expansion,arepresentationof the original differentialshouldbe
attainable if this istobe satisfied.Alsothroughuse of thisnumerical method,the truncationerror
associatedwitheachexpansioncanbe demonstrated.
0.0
1.0
2.0
3.0
0.0 0.5 1.0 1.5 2.0
Temperature
Grid point
Temperature (n)
Temp
Figure 1: Initial Conditions temperature step graph.

6. P a g e | 3
Theory
As computercapabilitieshave grown,sohave theirengineeringapplicationsandinaerodynamics,
the most notable of these isCFD.The methodsrise toprominence hassparkedthe engineering
communitytoinvestinitscomputingpowerinordertoutilise the benefitsonoffer.The opportunity
to testkey model platformswithoutanyoutlayforequipmentormaterials,otherthanthe program
itself,have made computational methodsakeyasset.Indesignorientatedprojectsthisformof
simulatedtestinghasbecome aparamountfeature,vastlyreducingcostsandincreasingthe
accuracy of the initial modelscreated.Parameterssuchas;flow speeds,pressure stagnation,wake
distributionandaerodynamicinefficienciescanbe monitoredandaddressedbefore production even
begins,inherentlyincreasingthe likelihoodof successof the openingmodel [1].
Partial Differentials
Partial differential equations(PDE’s) are inherentlydifficulttosolve astheycontainmore thanone
variable andinsteadare usedtodescribe problemsinvolvingthe parametersinuse whichcanthen
be solvedusingavariationof schemes. Theydescribe the rate of change of certainvariablesin
relationtoothers,whichwith thisassignmentbeingthe temperaturechangingwithtime (1st
term)
and the constantconvection velocityinthe 𝑥-direction,scaledwith the change between
temperature andstepsinthe 𝑥 direction(2nd
term):
𝜕𝑇
𝜕𝑡
+ 𝑈
𝜕𝑇
𝜕𝑥
= 0
Thisis a linear,one-dimensionalPDEasthe problemisonlyfirstorderandcontainsnothingraisedto
a higherpowerandall the functionsare transportedinone direction, 𝑥.Withthe initial boundary
conditionssupplied,the solutiontothisparticularpartial differential becomessimplyatransportof
itself alongthe time steps,demonstratedinthe example figure below:
Figure 2: Transport example of PDE function (University of Alaska, n.d.).
The 2nd
termin the PDE alsobares a certainresemblance toFourier’sLaw of Thermal Conduction:
𝑞 𝑥 = −𝑘
𝜕𝑇
𝜕𝑥
, Where 𝑘 is the materialsconductivity.
Thisillustratesthatitisinthis termthat the convectionpropertiesof the model are carried,
althoughinthiscase not by the, −𝑘, butby 𝑈, relatingtothe ConstantConvectionVelocityinthe 𝑥-
direction.
Although, thisisasimple problemandbefore the complex simulationprogramsbegantospawn,
more complex mathematical formulaehadtobe solved,withone methodbeing‘discretisation
schemes’,whichinitself isan umbrellatermcontainingamultitude of approaches thatshall be
alludedtointhe nextsection.

7. P a g e | 4
Discretisation Schemes
These methodsinclude;finite volume,approximatingthe volumearounddiscrete nodes,finite
element,the planeisdissectedintomanysmall elementsthatare approximated andmanymore,
but mostimportantlythe ‘finite-differenceapproximation’of interestinthispaper.
Discretisationisthe actof truncatingmodelsof continuousequationsintotheirsimpler,more finite
counterparts,allowingthemtobe appliedto andthenbe numericallydeduced,althoughtheystill
require significantcomputingpowerandthuswasseenasan earlyformof CFD [3]. The processof
the truncationintroducesaninherentinaccuracytothe model whichhasto be controlledand
monitoredthrough, ‘meshdiscretisation’,findingthe appropriate meshdensityorstepsize to
renderthe problemgridindependent,where increasingthe densityorreducingthe stepsize hasa
negligible effectof the accuracy of the yieldedanswer [4].
Finite-Difference Approximation
Finite-difference approximationsare of the mostderivativemethodsforsolvingdifferential
equations,postulatedbyMesserLeonhardEulerinthe 18th
century. Whenfacedwithequations
eitherverydifficulttosolve orisonlyapplicableinverystringent,finite situations,avariantmethod
of solutionisrequiredtogarneran answerthisiswhere the numerical processof finite-difference
approximationismostapplicable.The systemcanapproximate the solutionwiththe necessary
boundary andinitial conditionsimposed,providinganaccurate solutionforthe previous
unfathomable equation. Theyare of particularuse inaerodynamicsastheirtime andspace
dependentnature lendsitselftocomputingshockwave propagationorotherenergytransferflows.
Theirapproach usesthe act of ‘discretisation’toapproximate the differential, byapplyingafinite
grid,or mesh,of pointsat whichthe variablesare estimated,withthe processcontinuingasthe local
pointsgoverntheirapproximationvaluesfromthe neighbouringnodes.Iterative approximationin
thismannerproducesan obviouserror,knownasthe ‘discretisationortruncationerror’, diverging
fromthe true value.The keytothe principle is,like anything,minimisingthiserrorinthe system.
Monitoringthiserrorthenis somethingof paramountimportance andthroughthe implementation
of the ‘TaylorSeries’,thiscanbe achieved [5].
In additiontobeingeffectivelydefined,thereare three critical propertiesthatanyapproximationof
a partial differential shouldmaintain,being;consistency,stabilityandconvergence.The consistency
alludestothe finite-differencesaccuracyof approximationof the partial differential, whenthe delta
parameterstendto 0, thenthe approximationshouldconverge towardsthe true value of the
differential.Thisiswhere the use of the TaylorSeriesismostadvantageous,expandingeachtermat
the desiredpoint,enablingthe computationof the truncationerror.If the errorapproaches 0 as the
deltaparametersdosotoo, then itis saidto be consistent.Stability confirmsthatthe final solution’s
sensitivityiswithinthe acceptedrange andisn’ttoovolatile tosmall perturbationsinthe data.As
the data movesfromstepto step,the error can be compoundedand amplified;if thisisthe case
thenthe scheme isadducedto be unstable.Finally, convergence describesthe approximation
solutionsabilitytoiterate towardsthe original differentialsasthe meshdensitybecomesmore
sophisticated.Convergence demonstratesthatif the solutionisgridindependent,notinfluencedby
the meshdensity,thenthisisthe correctsolutiontothe original differential [6][7].
The three methodsunderscrutinyare all hyperbolicinnature whichdepictsa‘time dependent’
situationandtherefore all futuresolutionsare basedontheirpreviouscohortswithinthe region.

8. P a g e | 5
Anothersimilarityistheirexplicit characteristic,thisgovernsthatthe solutionateachgridpointis
givenbya concise formula.
Although,beforeinvestigatingthe finerpointsof the process,the three mostprominent
approximationmethodsshall be eludedto,beginningwiththe ‘Forward-Time,Backward-Space
Method’.
Forward-Time,Backward-SpaceScheme
Thisform of approximationmethodisa‘backward’,explicit,hyperbolicsystem, whichis where it
derivesitsname from.Thismeansthatthe nextsetof resultsare onlyderivedfromthe nodes
immediatelybehindthemgeometricallyinrelation totheirperviouscounterparts,asbecomes
apparentthroughinspectionof the equation, 𝑇𝑖
𝑛
− 𝑇𝑖−1
𝑛
:
𝑇𝑖
𝑛+1
−𝑇𝑖
𝑛
∆𝑡
+ 𝑈 {
𝑇𝑖
𝑛
−𝑇𝑖−1
𝑛
∆𝑥
} = 0
Thisalsogivesthe systemaninherentadvantage asthisencouragesconvergence,throughthe fact
that the approximationmethodhasa ‘domainof dependence’thatincludesthe initialdata,shared
by the partial differential.The domainof dependence statesthat the solutiontothe approximation
and the differentialattime 𝑡 = 0onlyappliesonthe initialdataset.
Furtherinformationgatheredfromthe equationitselfshowsthatthisisa firstordermethodand
mostsuitable tosimple differential approximations.The methodhasaregionof stabilityof 0 ≤ 𝑣 ≤
0.5, a hypothesisthatcanbe provenduringthe calculations.
Lax Scheme
Much like the previousisexplicitandhyperbolicinnature,yet,unlike itspredecessor,itisa ‘central
space’scheme,demonstratedthatall 𝑇 𝑛 termsare eitherside of 𝑖 andis firstorderaccurate for 𝑡,
althoughalsoencompassesasecondorderaccuracy for 𝑥, shownthroughinspectionof the right
handfraction:
𝑇𝑖
𝑛+1
−
𝑇 𝑖+1
𝑛 +𝑇 𝑖−1
𝑛
2
∆𝑡
+ 𝑈 {
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
} = 0
The previousdiscussedconditionsthatanapproximationmethodmustadhere toinorderto be
successful are summedupbyLax inthe ‘Lax Equivalence Theorem’,
‘For a well-posed linear, initial value problem with a consistent
discretisation, stability is the necessary and sufficient condition for
convergence of the numerical scheme’ (Strikwerda, 1989).
Thisstatesthat whena problemiscorrectlyconfinedwithappropriate initial andboundary
conditions,asmentionedinthe theory,andisconsistentandthusnow independentof the mesh
discretisation,thenforascheme tobe convergent,thenitmustalsobe stable.Thiscreates an
apparentinterdependency betweenthe three propertiesof approximationschemes. The stability
regionof Lax Methodis −0.5 ≤ 𝑣 ≤ 0.5.
Lax-WendroffScheme
Thisis the mostcomplicatedscheme of the three andmostrecent,itsderivationasrecentas1960 ,
yetstill hasthe hallmarksof beinghyperbolic andisexplicitwithtimesintegral andthusthe current
time stepistakenintoconsideration:

9. P a g e | 6
𝑇𝑖
𝑛+1
= 𝑇𝑖
𝑛
− 𝑈∆𝑡 {
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
} + 𝑈2 ∆𝑡2
2
{
𝑇𝑖+1
𝑛
−2𝑇𝑖
𝑛
+𝑇𝑖−1
𝑛
∆𝑥2
}
The addedterm, 2𝑇𝑖
𝑛
, inthe final bracketmovesthissystemawayfromthe previousLax Method
beinga centralisedscheme toasymmetrical method,beingsecondorderinboth 𝑥 and 𝑡. Thisis a
feature unique tothe Lax-Wendroff method,beingthe onlylinearadvectionmethodthat
encompassesthree nodesintoitscalculationsandalsobeingsecondorderaccurate. The stability
regionthissystemis the same as the Lax Methodpreviously.
Taylor Series
The Taylor Seriesisaform of evaluatingandrepresentingpartial differentials,althoughnot
exclusively,asaninfinite sumof itsterms ata single point,inthe formof seriesexpansion. There isa
special case of thissystem,knownasthe Maclaurin Series,namedafterthe Scottishmathematician
ColinMaclaurinwhodidextensive studyinthisarea,whenthe expansionisfocusedaround 0.The
methodwasoriginallyprovedbyJamesGregorywhenhe releasedhisbook ‘Vera Circuliet
HyperbolaeQuadratura’in1667.
The use of the serieshasmanyapplicationsinengineering,withitsmainbeingthe approximationof
functionsthroughthe expansiontothe necessarynumberof terms.Throughcollatingthe
appropriate numberof termsandthen‘truncating’the seriesavalidapproximationof the function
can be made.The act of truncatingthe seriesgeneratesanerror,althoughasthe expansion
continuesthe effectof eachtermdwindles,acharacteristicthatallowsthe truncationafteracertain
termnumber.The truncationerror can alsobe computedandgivesan indicationastothe validity
and performance of the initial approximationmade usingthe seriesexpansion.

10. P a g e | 7
Procedure
Havinga detailedoverviewof the procedural eventsof the experimentisof paramountimportance
and can jeopardise the validityof the resultsif thisisn’tupheld.A comprehensive guide enablesthe
conditionstobe recreatedif necessaryandresultsduplicated,eithersatisfyingtheirsoundnessor
throwingtheirclaimsintodisrepute.Withthisagendathe methodsof eachapproximationmethod
shall be alludedto.The approximationequationsfirsthave tobe manipulatedtomake 𝑇𝑖
𝑛+1
the
subjectandthen appliedtothe meshedgridtoyieldthe dataspread. The computational grid
employed21finite-difference gridpointsand12 timesstepsandthe excel equationsare specificto
calculatingthe cell 𝑛 + 1, 𝑖 + 2 inthe computationgrid,whichisdemonstratedinthe results,page
9.
Forward-Time, Backward-Space Scheme
Beginningwiththe firstapproximationmethodthe original equationhadtofirstbe rearrangedto
make the temperature the focus:
𝑇𝑖
𝑛+1
−𝑇𝑖
𝑛
∆𝑡
+ 𝑈 {
𝑇𝑖
𝑛
−𝑇𝑖−1
𝑛
∆𝑥
} = 0
=>
𝑇𝑖
𝑛+1
−𝑇𝑖
𝑛
∆𝑡
= −𝑈 {
𝑇𝑖
𝑛
−𝑇𝑖−1
𝑛
∆𝑥
}
=> 𝑇𝑖
𝑛+1
= 𝑇𝑖
𝑛
− 𝑈∆𝑡{
𝑇𝑖
𝑛
−𝑇𝑖−1
𝑛
∆𝑥
}
Withthe equationnowinthisform,the temperature atthe nextgeometricnode caneasilybe
calculatedthroughinputtingthe necessaryparameters, 𝑇𝑖
𝑛
representsthe stepimmediatelybefore
the one to be computedinthe gird,with 𝑇𝑖−1
𝑛
againin the previoustime step,butalsobackone grid
space also.It is thisapplicationpatternthatdemonstratesthe methodstitle of beinga‘backward
system’asall the grid pointsreferencedare backward.Beinginthe previoustimestepisanecessity,
itsthenbeing 𝑖 − 1 that givesthe systemitsname.
Deployingthisequationinexcel requiresthe predeterminedparameterstobe inputintocell blocks
readyto be selectedinthe requiredlocationwithinthe equation:
𝑇𝑖
𝑛+1
= 𝐶7 − (($𝐶$2∗ $𝐿$2)∗ ((𝐶7 − 𝐶6)/$𝐹$2))
$𝐶$2 = 𝑈, $𝐿$2 = ∆𝑡, $𝐹$2 = ∆𝑥.
In the excel equations,$appearsmixedinwiththe cell definition,employingthe symbolinthis way
fixesthe cell locationasthe equationisdraggedandcopiedacrossthe requiredcells,easingtheir
creation.
Withthe spreadsheetcreatedthe resultscanbe demonstrated,page 9:
Lax Scheme
Movingon to the lax scheme,the procedure isidentical to thatof the previousmethodandbegins
withthe algebraicmanipulation:
𝑇𝑖
𝑛+1
−
𝑇 𝑖+1
𝑛 +𝑇 𝑖−1
𝑛
2
∆𝑡
+ 𝑈 {
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
} = 0
=>
𝑇𝑖
𝑛+1
−
𝑇 𝑖+1
𝑛 +𝑇 𝑖−1
𝑛
2
∆𝑡
= −𝑈 {
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
}

11. P a g e | 8
=> 𝑇𝑖
𝑛+1
=
𝑇𝑖+1
𝑛
+𝑇𝑖−1
𝑛
2
− 𝑈∆𝑡{
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
}
The Lax Scheme isa centralisedmethodandtherefore,the previoustermof 𝑇𝑖
𝑛
isnot deployedin
thisscheme,insteadgridpointsabove andbelowthe locationbeingconsideredare usedtocalculate
itsvalue. 𝑇𝑖−1
𝑛
correspondstothe same grid locationas inthe previoussystemandnow the second
termthe equationdependsonis 𝑇𝑖+1
𝑛
,whichrelatestothe grid pointinfrontof the subjectlocation
inthe previoustime step.Thisexplainsthe centralisednature of the scheme,asthe subjectpointis
centredbetweenthe twothatinfluence it.
The Excel code for thisequationisslightlymore complicatedthanitspreviouscounterpartdue to
the centralisednature of the scheme:
𝑇𝑖
𝑛+1
= ((𝐶8 + 𝐶6)/2) − (($𝐶$2 ∗ $𝐿$2)∗ ((𝐶8 − 𝐶6)/(2 ∗ $𝐹$2)))
$𝐶$2 = 𝑈, $𝐿$2 = ∆𝑡, $𝐹$2 = ∆𝑥.
Again, the spreadsheetdepictingthe computational domainis illustratedon page 12.
Lax-Wendroff Scheme
Thisscheme hadalreadybeensetupwith 𝑇𝑖
𝑛+1
the subjectof the equationandthusneededno
manipulation:
𝑇𝑖
𝑛+1
= 𝑇𝑖
𝑛
− 𝑈∆𝑡 {
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
} + 𝑈2 ∆𝑡2
2
{
𝑇𝑖+1
𝑛
−2𝑇𝑖
𝑛
+𝑇𝑖−1
𝑛
∆𝑥2
}
The final method,Lax-Wendroff,isasymmetricscheme;the subjecttermiscalculatedusingthe
three adjacentgridpointsinthe previousstep,acombinationof the FT-BSandLax schemes.
Movingon againto the Excel code andthe lax-Wendroff methodisthe onlylinearapproximation
methodthatencompassesasymmetricsystem, using 3nodal positionstocompute the nextgrid
point:
𝑇𝑖
𝑛+1
= 𝐶7 − (($𝐶$2 ∗ $𝐿$2)∗ ((𝐶8 − 𝐶6)/(2 ∗ $𝐹$2)))+ ((($𝐶$2^2)∗
($𝐿$2^2)/2) ∗ ((𝐶8 − (2 ∗ 𝐶7) + 𝐶6)/($𝐹$2^2)))
$𝐶$2 = 𝑈, $𝐿$2 = ∆𝑡, $𝐹$2 = ∆𝑥.
Domaingrid,page 15.

12. P a g e | 9
Results
The three methods,afterbeinginputintoaMicrosoftExcel spreadsheetgenerateddatainboth
chart and tabularform.Depictingthe resultsinthiswayallowsforfurtheranalysistobe conducted
withgreaterease,identifyingeithertrendsorerrorsinthe method.Withthisinmind,the garnered
resultsfromthe Forward-Time,Backward-Space Scheme shall be illustratedfirst.
All resultsshall be demonstratedwiththe parameter 𝑣 =
𝑈∆𝑡
2∆𝑥
setto0.5, the theoretical upperlimit
of stability, whichinturn,throughmanipulation, setsavalue of ∆𝑡 = 0.1, witha sample calculation
fromthe initial conditions portrayed. Allcalculationswill be forthe gridpoint 𝑛 + 1, 𝑖 + 2.
Embodyingthisprocess,the Forward-Time,Backward-Space Scheme shall be employedfirst.
Forward-Time, Backward Space Scheme
Withthe equationalreadyadjustedandreadyforinputintothe grid, the computationsimply
requiresthe inputof the parameterdata:
𝑇𝑖
𝑛+1
= 𝑇𝑖
𝑛
− 𝑈∆𝑡{
𝑇𝑖
𝑛
−𝑇𝑖−1
𝑛
∆𝑥
}
=> 𝑇𝑖
𝑛+1
= 2 − 1(0.1) {
2−1
0.1
} = 1
Alongwitha calculusexample,the Excel inputcode isdepictedbelow,withthe relatingcells
highlighted:
𝑇𝑖
𝑛+1
= 𝐶7 − (($𝐶$2∗ $𝐿$2)∗ ((𝐶7 − 𝐶6)/$𝐹$2))
$𝐶$2 = 𝑈, $𝐿$2 = ∆𝑡, $𝐹$2 = ∆𝑥.
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 N+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 0 0 0 0 0 0 0 0 0 0 0
i+2 2 1 0 0 0 0 0 0 0 0 0 0
i+3 2 2 1 0 0 0 0 0 0 0 0 0
i+4 1 2 2 1 0 0 0 0 0 0 0 0
i+5 0 1 2 2 1 0 0 0 0 0 0 0
i+6 0 0 1 2 2 1 0 0 0 0 0 0
i+7 0 0 0 1 2 2 1 0 0 0 0 0
i+8 0 0 0 0 1 2 2 1 0 0 0 0
i+9 0 0 0 0 0 1 2 2 1 0 0 0
i+10 0 0 0 0 0 0 1 2 2 1 0 0
i+11 0 0 0 0 0 0 0 1 2 2 1 0
i+12 0 0 0 0 0 0 0 0 1 2 2 1
i+13 0 0 0 0 0 0 0 0 0 1 2 2
i+14 0 0 0 0 0 0 0 0 0 0 1 2
i+15 0 0 0 0 0 0 0 0 0 0 0 1
i+16 0 0 0 0 0 0 0 0 0 0 0 0
i+17 0 0 0 0 0 0 0 0 0 0 0 0
i+18 0 0 0 0 0 0 0 0 0 0 0 0
i+19 0 0 0 0 0 0 0 0 0 0 0 0
i+20 0 0 0 0 0 0 0 0 0 0 0 0
Table 1: Forward-Time, Backward-Space computational grid (𝒗 = 𝟎. 𝟓).

13. P a g e | 10
Usingthe data calculatedinthe computational grid,graphsateachtime stepcan be plotted
illustratingthe functions transcendence througheachtime step:
Figure 3: Forward-Time, Backward-Space Scheme step graphs. (𝒗 = 𝟎. 𝟓)
The resultsobtainedare with 𝑣 = 0.5,inthe nextsimulation thisvalue hasbeenadjustedto 𝑣 =
0.51, whichisnowabove the theoretical stabilityvalueandshoulddemonstrate the method
becomingunstable,usingthe stepsof 𝑛 + 2, 𝑛 + 6 and 𝑛 + 11 to illustratesthe progressionthrough
the simulation:
Figure 4: Forward-Time, Backward-Space Scheme step graphs. (𝒗 = 𝟎. 𝟓𝟏)
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n
n
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+1
n+1
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+2
n+2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+3
n+3
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+4
n+4
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+5
n+5
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+6
n+6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+7
n+7
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+ 8
n+ 8
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+ 9
n+ 9
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 10
n + 10
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+2
n+2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+6
n+6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11

14. P a g e | 11
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 -0.02 0.0004 -8e-6
1.6e-7
-3.2e-9
6.4e-11
-1.3e-12
2.56e-14
-5.1e-16
1.02e-17
-2e-19
i+2 2 0.98 -0.04 0.0012 -3.2e-5
8.1e-7
-1.9e-8
4.54e-10
-1e-11
2.34e-13
-5.2e-15
1.14e-16
i+3 2 2 0.9596 -0.0599 0.0024 -8.2e-5
2.46e-6
-6.9e-8
1.84e-9
-4.7e-11
1.19e-12
-2.9e-14
Table 2: Forward-Time, Backward-Space Scheme computational grid. (𝒗 = 𝟎. 𝟓𝟏)
Onlycertainstepshave been depictedasthisisa demonstrationof the stabilityboundary.The same
procedure of demonstratingthe stabilitylimitshall now be conductedonthe lowerconstraintvalue,
usingthe same steplocalesof before. The theoretical lowerlimitis 𝑣 = 0:
Figure 5: Forward-Time, Backward-Space Scheme step graphs. (𝒗 = 𝟎)
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 1 1 1 1 1 1 1 1 1 1 1
i+2 2 2 2 2 2 2 2 2 2 2 2 2
i+3 2 2 2 2 2 2 2 2 2 2 2 2
Table 3: Forward-Time, Backward-Space Scheme computational grid. (𝒗 = 𝟎)
Thenadjustingthe parametertomove beyondthe limit, 𝑣 = −0.01,demonstratesthe previous
examples significance:
Figure 6: Forward-Time, Backward-Space Scheme step graphs. (𝒗 = −𝟎. 𝟎𝟏)
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 1.02 1.0404 1.0612 1.0824 1.1041 1.1261 1.1487 1.1717 1.1951 1.2189 1.2434
i+2 2 2.02 2.04 2.0599 2.0799 2.0999 2.1198 2.1397 2.1595 2.1793 2.1989 2.2186
i+3 2 2 1.9996 1.9987 1.9976 1.9959 1.9938 1.9913 1.9883 1.9849 1.9810 1.9767
Table 4: Forward-Time, Backward-Space Scheme computational grid. (𝒗 = −𝟎. 𝟎𝟏)
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+2
n+2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+6
n+6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+2
n+2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n+6
n+6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11

15. P a g e | 12
Lax Scheme
Again, beginningwiththe previouslymanipulatedequationandinputtingthe selecteddata:
𝑇𝑖
𝑛+1
=
𝑇𝑖+1
𝑛
+𝑇𝑖−1
𝑛
2
− 𝑈∆𝑡{
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
}
=> 𝑇𝑖
𝑛+1
=
2+1
2
− 1(0.1) {
2−1
2(0.1)
} = 1
The Excel code for thisequationagainhighlightsthe cellsused:
𝑇𝑖
𝑛+1
= ((𝐶8 + 𝐶6)/2) − (($𝐶$2 ∗ $𝐿$2)∗ ((𝐶8 − 𝐶6)/(2 ∗ $𝐹$2)))
$𝐶$2 = 𝑈, $𝐿$2 = ∆𝑡, $𝐹$2 = ∆𝑥.
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 0 0 0 0 0 0 0 0 0 0 0
i+2 2 1 0 0 0 0 0 0 0 0 0 0
i+3 2 2 1 0 0 0 0 0 0 0 0 0
i+4 1 2 2 1 0 0 0 0 0 0 0 0
i+5 0 1 2 2 1 0 0 0 0 0 0 0
i+6 0 0 1 2 2 1 0 0 0 0 0 0
i+7 0 0 0 1 2 2 1 0 0 0 0 0
i+8 0 0 0 0 1 2 2 1 0 0 0 0
i+9 0 0 0 0 0 1 2 2 1 0 0 0
i+10 0 0 0 0 0 0 1 2 2 1 0 0
i+11 0 0 0 0 0 0 0 1 2 2 1 0
i+12 0 0 0 0 0 0 0 0 1 2 2 1
i+13 0 0 0 0 0 0 0 0 0 1 2 2
i+14 0 0 0 0 0 0 0 0 0 0 1 2
i+15 0 0 0 0 0 0 0 0 0 0 0 1
i+16 0 0 0 0 0 0 0 0 0 0 0 0
i+17 0 0 0 0 0 0 0 0 0 0 0 0
i+18 0 0 0 0 0 0 0 0 0 0 0 0
i+19 0 0 0 0 0 0 0 0 0 0 0 0
i+20 0 0 0 0 0 0 0 0 0 0 0 0
Table 5: Lax Scheme computational grid (𝒗 = 𝟎. 𝟓).
Thisproducedstepgraphswhichcan be seen below asfigure 7:

16. P a g e | 13
Figure 7: Lax Scheme step graphs (𝒗 = 𝟎. 𝟓).
Again,demonstratingthe upperlimitof stabilityfirst, 𝑣 = 0.51:
Figure 8: Lax Scheme step graphs (𝒗 = 𝟎. 𝟓𝟏).
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 -0.02 -0.0099 0.0004 0.0002 -1e-5
-5e-6
2.87e-7
1.4e-7
-8.7e-9
-4.2e-9
2.76e-10
i+2 2 0.99 -0.0403 -0.0198 0.0010 0.0005 -2.9e-05
-1.4e-5
8.69e-7
4.23e-7
-2.8e-8
-1.3e-8
i+3 2 2.01 0.9797 -0.0609 -0.0297 0.0018 0.0009 -5.8e-5
-2.8e-5
1.88e-6
9.12e-7
-6.3e-8
Table 6: Lax Scheme computational grid (𝒗 = 𝟎. 𝟓𝟏).
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n
n
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 1
n + 1
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 3
n + 3
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 4
n + 4
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 5
n + 5
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 7
n + 7
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 8
n + 8
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 9
n + 9
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 10
n + 10
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11

17. P a g e | 14
Withnow provingthe lowerboundarypoint,fromthe theory, 𝑣 = −0.5,asthe value migrationsare
so small these figuresshall be supportedwiththe first4rows of the computational grid to
demonstrate the changesclearly:
Figure 9: Lax Scheme step graphs (𝒗 = −𝟎. 𝟓).
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 2 2 1 0 0 0 0 0 0 0 0
i+2 2 2 1 0 0 0 0 0 0 0 0 0
i+3 2 1 0 0 0 0 0 0 0 0 0 0
Table 7: Lax Scheme computational grid (𝒗 = −𝟎. 𝟓).
Movingbeyondthislimit, 𝑣 = −0.51:
Figure 10: Lax Scheme step graphs (𝒗 = −𝟎. 𝟓𝟏).
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 2.02 2.0301 0.9894 -0.0616 -0.0406 0.0019 0.0014 -5.9e-5
-5e-5
1.91e-6
1.74e-6
i+2 2 2.01 0.9797 -0.061 -0.0402 0.0018 0.0014 -5.8e-5
-5e-5
1.89e-6
1.72e-6
-6.3e-8
i+3 2 0.99 -0.040 -0.03 0.0012 0.0010 -3.9e-5
-3.5e-5
1.29e-6
1.21e-6
-4.3e-8
-4.3e-8
Table 8: Lax Scheme computational grid (𝒗 = −𝟎. 𝟓𝟏).
A final areaof interestiswhen 𝑣 = 0,as thisthensets ∆𝑡 = 0 too:
Figure 11: Lax Scheme step graphs (𝒗 = 𝟎).
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11

18. P a g e | 15
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 1 0.75 0.625 0.5 0.4375 0.3594 0.3281 0.2734 0.2578 0.2168 0.2095
i+2 2 1.5 1.25 1 0.875 0.7188 0.6563 0.5469 0.5156 0.4336 0.4189 0.3545
i+3 2 1.5 1.25 1.125 0.9375 0.875 0.7344 0.7031 0.5938 0.5801 0.4922 0.4888
Table 9: Lax Scheme computational grid (𝒗 = 𝟎).
Lax-Wendroff Scheme
The final methods resultstodisplayare the Lax-Wendroff method,byfarthe mostcomplicatedpf
the three:
𝑇𝑖
𝑛+1
= 𝑇𝑖
𝑛
− 𝑈∆𝑡 {
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
} + 𝑈2 ∆𝑡2
2
{
𝑇𝑖+1
𝑛
−2𝑇𝑖
𝑛
+𝑇𝑖−1
𝑛
∆𝑥2
}
=> 𝑇𝑖
𝑛+1
= 2 − (1)(0.1){
2−1
2(0.1)
} + 12 0.12
2
{
2−2(1)+1
0.12
} = 1
Excel code:
𝑇𝑖
𝑛+1
= 𝐶7 − (($𝐶$2 ∗ $𝐿$2)∗ ((𝐶8 − 𝐶6)/(2 ∗ $𝐹$2)))+ ((($𝐶$2^2)∗
($𝐿$2^2)/2) ∗ ((𝐶8 − (2 ∗ 𝐶7) + 𝐶6)/($𝐹$2^2)))
$𝐶$2 = 𝑈, $𝐿$2 = ∆𝑡, $𝐹$2 = ∆𝑥.
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 0 0 0 0 0 0 0 0 0 0 0
i+2 2 1 0 0 0 0 0 0 0 0 0 0
i+3 2 2 1 0 0 0 0 0 0 0 0 0
i+4 1 2 2 1 0 0 0 0 0 0 0 0
i+5 0 1 2 2 1 0 0 0 0 0 0 0
i+6 0 0 1 2 2 1 0 0 0 0 0 0
i+7 0 0 0 1 2 2 1 0 0 0 0 0
i+8 0 0 0 0 1 2 2 1 0 0 0 0
i+9 0 0 0 0 0 1 2 2 1 0 0 0
i+10 0 0 0 0 0 0 1 2 2 1 0 0
i+11 0 0 0 0 0 0 0 1 2 2 1 0
i+12 0 0 0 0 0 0 0 0 1 2 2 1
i+13 0 0 0 0 0 0 0 0 0 1 2 2
i+14 0 0 0 0 0 0 0 0 0 0 1 2
i+15 0 0 0 0 0 0 0 0 0 0 0 1
i+16 0 0 0 0 0 0 0 0 0 0 0 0
i+17 0 0 0 0 0 0 0 0 0 0 0 0
i+18 0 0 0 0 0 0 0 0 0 0 0 0
i+19 0 0 0 0 0 0 0 0 0 0 0 0
i+20 0 0 0 0 0 0 0 0 0 0 0 0
Table 10: Lax-Wendroff Scheme computational grid (𝒗 = 𝟎. 𝟓).
Finally,withthe computationgridcompletethe necessarystepgraphsof the Lax-Wendroff method
can be produced(figure 11):

19. P a g e | 16
Figure 12: Lax-Wendroff Scheme step graphs (𝒗 = 𝟎. 𝟓).
Finally, againproceedingtoprove the stabilityboundaries,whichinthissystemare the same
boundaryvaluesasthe Lax Scheme,therefore beginningwiththe upperlimit,𝑣 = 0.51:
Figure 13: Lax-Wendroff Scheme step graphs (𝒗 = 𝟎. 𝟓𝟏).
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 -0.02 0.0107 -0.0008 0.0003 -3.5e-5
8.69e-6
-1.5e-6
3.44e-7
-6.7e-8
1.5e-8
-3.1e-9
i+2 2 0.9698 -0.0395 0.0222 -0.0024 0.0007 -0.0001 2.78e-5
-5.2e-6
1.21e-6
-2.5e-7
5.61e-8
i+3 2 1.9898 0.9393 -0.0584 0.0345 -0.0046 0.0014 -0.0003 6.31e-5
-1.3e-5
3e-06
-6.4e-7
Table 11: Lax-Wendroff Scheme computational grid (𝒗 = 𝟎. 𝟓𝟏).
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n
n
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 1
n + 1
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 3
n + 3
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 4
n + 4
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 5
n + 5
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 7
n + 7
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 8
n + 8
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 9
n + 9
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 10
n + 10
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11

20. P a g e | 17
Finally,againproceedingtoprove the lowerstabilityboundary,at 𝑣 = −0.5:
Figure 14: Lax-Wendroff Scheme step graphs (𝒗 = −𝟎. 𝟓𝟎).
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 2 2 1 0 0 0 0 0 0 0 0
i+2 2 2 1 0 0 0 0 0 0 0 0 0
i+3 2 1 0 0 0 0 0 0 0 0 0 0
Table 12: Lax-Wendroff Scheme computational grid (𝒗 = −𝟎. 𝟓𝟎).
Thenexceedingthe stabilityboundary, 𝑣 = −0.51:
Figure 15: Lax-Wendroff Scheme step graphs (𝒗 = −𝟎. 𝟓𝟏).
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 2.02 1.9682 0.8881 -0.0962 0.0507 -0.0090 0.0027 -0.0006 0.0001 -3.1e-5
7.51e-6
i+2 2 1.9898 0.9393 -0.0586 0.0454 -0.0068 0.0023 -0.0004 0.0001 -2.5e-5
6.06e-6
-1.4e-6
i+3 2 0.9698 -0.0395 0.0330 -0.0039 0.0014 -0.0003 6.84e-5
-1.4e-5
3.51e-6
-7.7e-7
1.85e-7
Table 13: Lax-Wendroff Scheme computational grid (𝒗 = −𝟎. 𝟓𝟏).
The finallyatwhen 𝑣 = 0:
Figure 16: Lax-Wendroff Scheme step graphs (𝒗 = 𝟎).
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 2
n + 2
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 6
n + 6
-0.5
0.5
1.5
2.5
1 3 5 7 9 11 13 15 17 19 21
n + 11
n + 11

21. P a g e | 18
n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11
i 0 0 0 0 0 0 0 0 0 0 0 0
i+1 1 1 1 1 1 1 1 1 1 1 1 1
i+2 2 2 2 2 2 2 2 2 2 2 2 2
i+3 2 2 2 2 2 2 2 2 2 2 2 2
Table 14: Lax-Wendroff Scheme computational grid (𝒗 = 𝟎).

22. P a g e | 19
Analysis
Throughan in-depthanalysisof the data,laterjudgementsonall three methodsusedcanbe made,
dissectingtheirstrengthsandweaknessandalludingtohow theycan be improvedmovingforwards.
Throughthe initial datasetsof eachapproximationmethoditbecomesapparenttosee thatonthe
upperboundaryof stability,when 𝑣 = 0.5, creatingatime step, ∆𝑡 = 0.1, theyall accurately
representthe partial differential overthe time stepsrequested.The graphsshow nodistortionaway
fromthe initial differentialssolution,theyare simplytransportedalongthe time steps.Thisiswhere
the comparative similarities betweeneachmethodbegintodiverge andthe differencesbecome
more noticeable ineachsystem.
Thisbeginswithwhen 𝑣 issetto 0, The Forward-Time,Backward-Space system, inwhichthisvalueis
the theoretical lowerlimitof stability,demonstrates the initialtime stepgraphdepictedinthe
objectives, Figure 1,although,insteadof transportingthisshape througheachtime step,aswhen
setto its upperstabilityboundary,the graphremainsstationaryrepeatingitself overthe perceived
time window. The Lax-Wendroff methodbehavesthe same manner,recyclingthe stepsize asthe
time progresses,althoughforthismethodscase,thisisn’tthe lowerstabilityvalue forthe system.
The Lax methoddiffersinthisscenario,againthis notlinkedtoitsstabilityvalue limitsandiswell
withinthe regionandthe graphagain remainsstationaryanddoesn’ttranspose throughthe steps,
yetthe approximationshowssignsof inaccuracies,withthe predictionerroramplifyingthroughthe
windowasthe graph decays, Figure 11. Whilstthe othertwomethodremainconstant,the Lax
exhibitsadropinpeakpredictionvalue from2in 𝑛, to 0.4888 whenat 𝑛 + 11, illustratingan error
of 75.56% fromthe true value of the differentialatthisstage. The decayrate appearsto be sub
linearandan error analysisshall be conductedtovalidate thishypothesis.
Whenthenmovedtoset, 𝑣 to the lowerlimitof the followingtwomethods, −0.5, ∆𝑡 = −0.1,they
bothbehave inthe same manner.Theydemonstrate the peakvalue of the partial differential inthe
initial steps,untiltheythe systemspredicts 0 overthe restof the testedwindow. Whensettothe
positive limitboundthe predictionmovesforwards,fromthe toplefttothe bottomrightof the
computational domain,bothforwards,intime andingridpoints.Yetnow inthissetup,it becomes
apparentinthe fewperceivablestepsbefore the gridbecomesall zerosthatnow the valuesare
translatingperpendicularlytothis,e.g.frombottomlefttotopright.
Thus far,these resultshave stayedwithinthe theoretical boundsof the methodsstability as
depictedinthe theory.Thislimitcanonlybe verifiedbydemonstratingthe effectsof passing
throughit,whichas, whenall setto the upper value of, 𝑣 = 0.51,the graphsshow earlysignsof an
instability,illustratedthroughthe growingamplitude inthe oscillationsof the value asitfluctuates
betweenpositiveandnegativesigns (FT-BS- Figure 4,Lax - Figure 8, L-W - Figure 13). There is
furtherevidence of thisinthe computational gridsasthe magnitude onthe value,bothpositive and
negative,isrisingasthe systemmovesthroughthe time andgridsteps,diagonallydowntothe right.
The patternof these fluctuationsalsofollowsa distinctpatternfrommethodtomethod,withthe
FT-BSand Lax-Wendroff Schemesbothmaintainingaswitchinsignaftereach step,yetthe Lax
Scheme changeseverytwostepsalongthe computationalgrid(FT-BS- Table 2, Lax - Table 6, L-W -
Table 11). In additiontothe signvolatility,the overpredictionlevelof the eachmethodrisesandin
line withthe 𝑣 = 0 analysis,anerrorshall be plottedanddiscussedlatertoallude toanypatternin
the data.

23. P a g e | 20
Movingtowardsexceedingthe negativeboundary, 𝑣 = −0.51,where ∆𝑡 = −0.102 andthe data
beginstobehave ina combinationof when at 𝑣 = 0.51, as the data isunstable andfollowingthe
same trendsas before andwhenat −0.5,as the directionof migrationof the resultshasagain
movedthrough 90° anti-clockwise,perpendiculartowhenthe value ispositive.The valuesare
growinginsize as theymove throughthe steps,signsof anunstable method. The majordifference
betweeneachlimitisthe magnitude of the values,whenatthe positive boundtheyare still inthe
same orderof magnitude tothe true answer,yeton the negative bound, theyare considerably
lower,tothe pointthat theirmovementscanonlybe witnessedinthe dataanddon’tappearon the
standardscale usedfor the restof the graphs.

24. P a g e | 21
Discussion
From use of the theory,the measurementof success whendeployingafinite-difference
approximationmethodisgaugedinthree majorcategories:
 Consistency
 Stability
 Convergence
Throughdiscussingthe variousmethodsinaccordance withthese classifications,thentheirrelative
performance toone anothercan be yieldedandconclusionsdrawn.
Beginningwiththe consistency,fromthe theory:
The consistency alludes to the finite-differences accuracy of approximation of the partial
differential, when the delta parameters tend to 0, then the approximation should converge
towards the true value of the differential.
Thisis alsowhere the use of a Taylor seriesexpansioncanhelptodemonstrate amethods stability,
as throughits implementation,the original PDEshouldbe attainable if the methodisconsistent.
Therefore takingthe systemsinturn:
Forward-Time, Backward-Space Taylor Series Expansion
𝑇𝑖
𝑛+1
−𝑇𝑖
𝑛
∆𝑡
+ 𝑈 {
𝑇𝑖
𝑛
−𝑇𝑖−1
𝑛
∆𝑥
} = 0
InsertTaylorSeriesandrearrange to give:
𝜕𝑇
𝜕𝑡
+ 𝑈
𝜕𝑇
𝜕𝑥
+
∆𝑡
2!
𝜕2 𝑇
𝜕𝑡2
− 𝑈
∆𝑥
2!
𝜕2 𝑇
𝜕𝑥2
+ ⋯ = 0
When∆𝑡 → 0 and ∆𝑥 → 0:
𝜕𝑇
𝜕𝑡
+ 𝑈
𝜕𝑇
𝜕𝑥
= 0
Thissatisfies the methodsstatementof tendingthe deltaparametersto0 and indoingso the
methodisproventobe consistentasthe final equationmirrorsthatof the original PDE.
FT-BS TruncationError
The truncationerror relatestothe termsthat are approximatedtozerowhenattemptingtoprove
consistency,the termsof the mostsignificance are selected,whichinitself makesthe truncation
error value anapproximationalso:
𝑇. 𝐸 =
∆𝑡
2!
𝜕2 𝑇
𝜕 𝑡2
− 𝑈
∆𝑥
2!
𝜕2 𝑇
𝜕𝑥2
+ ⋯
Lax Scheme Taylor Series Expansion
𝑇𝑖
𝑛+1
−
𝑇 𝑖+1
𝑛 +𝑇 𝑖−1
𝑛
2
∆𝑡
+ 𝑈 {
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
} = 0
InsertTaylorSeriesandrearrange to give:
𝜕𝑇
𝜕𝑡
+ 𝑈
𝜕𝑇
𝜕𝑥
+
∆𝑡
2!
𝜕2 𝑇
𝜕𝑡2
−
∆𝑥
2∆𝑡
𝜕2 𝑇
𝜕𝑥2
+ 𝑈
∆𝑥2
3!
𝜕3 𝑇
𝜕𝑥3
… = 0
When∆𝑡 → 0 and ∆𝑥 → 0:

25. P a g e | 22
𝜕𝑇
𝜕𝑡
+ 𝑈
𝜕𝑇
𝜕𝑥
−
∆𝑥2
2∆𝑡
𝜕2 𝑇
𝜕𝑥2
= 0
In thiscase again the equationdiffersfromthe original andisthereforedeemedinconsistent.
Lax TruncationError
𝑇. 𝐸 = 𝑈
∆𝑥2
3!
𝜕3 𝑇
𝜕𝑥3
…
Lax-Wendroff Scheme Taylor Series Expansion
𝑇𝑖
𝑛+1
= 𝑇𝑖
𝑛
− 𝑈∆𝑡 {
𝑇𝑖+1
𝑛
−𝑇𝑖−1
𝑛
2∆𝑥
} + 𝑈2 ∆𝑡2
2
{
𝑇𝑖+1
𝑛
−2𝑇𝑖
𝑛
+𝑇𝑖−1
𝑛
∆𝑥2
}
Due to the complexityof the final method,aslightlydifferentapproachisneededto
comprehensivelyexpandthe scheme,beginningwith anexpansionintime truncatedafterthe
secondderivative:
𝑇𝑖
𝑛+1
≈ 𝑇𝑖
𝑛
+ ∆𝑡
𝜕𝑇
𝜕𝑡
+
∆𝑡2
2!
𝜕2 𝑇
𝜕𝑡2
+ ⋯
The proceedingforwardrequiresthe introductionof the equationtobe solved,whichreplacesthe
firstderivative intime:
𝜕𝑇
𝜕𝑡
= −𝑈
𝜕𝑇
𝜕𝑥
Thenthroughdifferentiatingthisequation,the secondderivativeinthe original expansioncanalso
be replaced:
𝜕2 𝑇
𝜕 𝑡2
= −𝐴
𝜕
𝜕𝑡
(
𝜕𝑇
𝜕𝑥
) = −𝐴
𝜕
𝜕𝑥
(
𝜕𝑇
𝜕𝑡
) = 𝐴2 𝜕2 𝑇
𝜕𝑥2
YieldingTaylorSeriesintime of:
𝑇𝑖
𝑛+1
= 𝑇𝑖
𝑛
− 𝐴∆𝑡
𝜕𝑇
𝜕𝑡
+ 𝐴2∆𝑡2
2!
𝜕2 𝑇
𝜕𝑥2
+ ⋯
The final taskis to approximate the spatial derivativesusingthe centre-difference formula:
𝑇𝑖
𝑛+1
= 𝑇𝑖
𝑛
− (
𝐴∆𝑡
2∆𝑥
)( 𝑇𝑖+1
𝑛
− 𝑇𝑖−1
𝑛 )+ 2(
𝐴∆𝑡
2𝑥
)
2
( 𝑇𝑖+1
𝑛
− 2𝑇𝑖
𝑛
+ 𝑇𝑖−1
𝑛 )
Thisdemonstratesthatthe Lax-Wendroff,likethe Forward-Time,Backward-Space Scheme is
consistentwiththe original PDE.
Lax-WendroffTruncationError
𝑇. 𝐸 =
∆𝑡2
2!
𝜕2 𝑇
𝜕 𝑡2
+
∆𝑥2
2!
𝜕2 𝑇
𝜕𝑥2
+ ⋯

26. P a g e | 23
Withthe consistencynowestablishedforeachmethod,the stabilitycanbe investigated,whichwas
achievedthroughthe variationof the 𝑣 parameter,whichinturnadjustedthe time stepof the grid.
Variationinthismannerhelpstoallude tohow the systemsreactto the computationdomainbeing
altered,changingthe meshdiscretisationof the problem. The stabilityfromthe theorydictates:
Stability confirmsthatthefinal solution’ssensitivity is within the accepted rangeand isn’t too
volatile to small perturbations in the data.
Throughthe analysisandthe resultsgarneredformperformingthe methods,itbecomesapparent
that the Forward-Time,Backward-Space Scheme,althoughconsistent,isn’tstable assmall
movementsin ∆𝑡,hasprofoundeffectsonthe errorof the peakvalue,demonstratedinthe tables
below whichembodyaincrementvariationof 0.02:
Δt = 0.08 Δt = 0.06 Δt = 0.04 Δt = 0.02
Step Error (%) Step Error (%) Step Error (%) Step Error (%)
n 0 n 0 n 0 n 0
n+1 20 n+1 40 n+1 60 n+1 80
n+2 29.6 n+2 56.8 n+2 79.2 n+2 94.4
n+3 38.56 n+3 69.76 n+3 89.76 n+3 98.56
n+4 46.75 n+4 79.26 n+4 95.14 n+4 99.65
n+5 54.12 n+5 86.00 n+5 97.75 n+5 99.92
n+6 60.68 n+6 90.67 n+6 98.98 n+6 99.98
n+7 60.68 n+7 90.67 n+7 98.98 n+7 99.98
n+8 66.45 n+8 93.84 n+8 99.54 n+8 99.99
n+9 71.48 n+9 95.97 n+9 99.79 n+9 99.99
n+10 75.84 n+10 97.38 n+10 99.91 n+10 99.99
n+11 79.59 n+11 98.30 n+11 99.96 n+11 99.99
Table 15: Time step variation FT-BS.
Thisclearlyillustratesthe instabilityof thismethod,showinghow sensitiveitistosmall changesin
the time step.
Movingon to the Lax method,The graphsin the resultsshow how muchmore stable itisin
comparisontothe FT-BSstill demonstratingthe convectionthroughthe time stepthroughoutthe
whole range of valuesfor ∆𝑡. Thiscoupledwithitbeinginconsistent yetalleviatestoamethodthat
encompassesthe necessarycriteriatobe a successful approximationmethod. Furtherempirical
evidence of the Lax Methodsinconsistencyisthatwhen 𝑣 = 0,the graphs shouldhave
demonstratedthe initial conditions,somethingthe Forward-Time,Backward-Space andLax-
Wendroff schemesbothsatisfied,whereasthe Lax scheme continuedtoapproximatestepsmoving
forwards.The followinggraphillustratesthe percentageerrorincomparisontoif the stepswere still
taken:

27. P a g e | 24
Figure 17: Percentage error of Lax Method (𝒗 = 𝟎).
The graph demonstratesasub-linearrate of convergence inrelationtoerrorasit grows,reinforcing
the consistencyclaim.
Finallythe Lax-Wendroff Scheme,the mostcomplex of the three andhasalreadybeenproventobe
constantwiththe original PDE.Thisagain thoughthe data spreadinthe resultscan be seenas
stable,withinitsstabilitylimitsitcanaccuratelypredictthe differential.Whenall three stepover
theirstabilityboundaryat 𝑣 = 0.51,witha time stepof ∆𝑡 = 0.102, the Lax-Wendroffisbyfar the
mostcapable at maintainaccuracy as illustrated infigure 18,where all three methodsare
demonstratedonthe same graphto ease directcomparisonstobe made:
Figure 18: Relative overshoot error percentage (𝒗 = 𝟎. 𝟓𝟏).
As the graph showsthe othertwomethodsfarexceedthe errorof the Lax-Wendroff Scheme,which
showssignsof the error growthrate reducingasmore time stepsare taken.
The final criteriontosatisfyisthe convergence:
The approximation solutions ability to iterate towards the original differentials as the mesh
density becomes more sophisticated.
0
10
20
30
40
50
60
70
80
0 5 10 15
PercentageError(%)
Time Step (n + )
Percentage Error of Lax Method (V=0)
% error
0
2
4
6
8
10
12
0 5 10 15
PercentageError(%)
Time Step (n+)
Relative Overshoot Error (V=0.51)
FT-BS
Lax
LW

28. P a g e | 25
It isalso knownfrom Strikwerdathat,‘stabilityisthe necessaryandsufficientconditionfor
convergence of the numerical scheme.’Whichimmediatelydeniesthe Forward-Time,Backward-
Space Scheme thisaccreditation.The Lax-Wendroff certainlyachievesthisdirective,aswhenthe
stepsize isaltered,the residual errorinducesis withinanacceptable regioncomparedtothe other
methodstobe describedasconvergentandthe problembecomesgridindependentwhen ∆𝑡 = 0.1.
It became clearduringthe analysisthatwhenthe twoLax methodsproceededtovaluesof 𝑣 that
were negative, the patternsinthe dataalteredslightly.Thiscanbe adjudgedtobe the effectof then
employinganegative timestep,insteadof the datamigratingdiagonallydownandright,thischange
of signresultedinthe migrationmovingupinstead,whichiswhy there are veryfew non-zero
numbersinthe computational grid,the gridpointbeingusedtocalculate the nexthasbeenreversed
and istherefore takingintoaccountmanymore zeros.
It was alsohighlightedthatthe patternof signmigrationwasslightly differentforthe Lax method,as
the signreverse aftereverytwostepsratherthaneveryone forthe othertwo methods.Thiscanbe
attributedtothe primaryequationusedtocalculate the gridpoint.The Lax doesn’tencompassa 𝑇𝑖
𝑛
termand as resultsfeelsthe effectof the signchange a steplaterthanthe other twomethods,
hence the delayedsignchange.

29. P a g e | 26
Conclusion
In conclusion,withall the evidence supplied,itiscleartosee the Lax-Wendroff isthe bestscheme in
termsof accuracy, yetit isalsothe most complex andthe additional computingtime isnotalways
goingto be an efficient use of time.The secondbestwouldbe the Forward-Time,Backward-Space
Scheme,asalthoughmore crude than the Lax Method,inmore situations itmirrorthe original PDE,
yetlostaccuracy the mostreadilyundervariationsinthe time step.
The processmust be seenas a successas the resultsmirroredthatof the theoretical information
highlightedinthe earlystagesof thispaper.All the objectiveshave beensatisfiedandwiththe
inherentcohesive nature of the datawiththe theoretical,resultvalidityandreliabilityclaimsare
refuted.Throughvariationof the time stepthe methodswereconclusivelytestedandthe three
criteriathat demonstrate finite-difference approximationmethodsvaliditystudied.Throughuse of
TaylorSeriesExpansionsthe constancy of eachwasillustrated,beforeinvestigatingthe stabilityand
convergence throughvaryingthe time step.
Improvingthisprocessif more time wasaffordedwouldfurtherenhance the resultsstature and
couldbe achievedbysettingupthe spreadsheetsagainseparatelytodial outanyhuman errorwith
data input.Also,byapplyingthe approximationschemestoa range of partial differentialswould
helptofurtherratifythe superiorityof agivensystem.

30. P a g e | 27
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