Algorithm Analysis of Numerical Solutions to the Heat Equation

Transcription

1 Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober Algorihm Analysis of Numerical Soluions o he Hea Equaion Edmund Agyeman Deparmen of Mahemaics, Kwame Nkrumah Universiy of Science and Technology, Kumasi-Ghana. Derick Folson Deparmen of Mahemaics, Kwame Nkrumah Universiy of Science and Technology, Kumasi-Ghana. ABSTRACT The numerical algorihms employed in he soluion of Parabolic Parial Differenial Equaions are he subjec of his paper. In paricular, he Crank-Nicolson scheme, which is generally acceped as an improvemen of he Schmid scheme, is subjeced no only o sabiliy analysis, bu also absolue relaive error analysis o guide Mahemaicians and Engineers alike o know he rue performance of hese numerical soluion mehods. The Hea Equaion wih Dirichle condiions conducing hea is analysed by employing he analyical mehod of soluion where he mehod of Separaion of Variables is used. The same equaion is hen solved wih he Schmid scheme as well as he Crank-Nicolson scheme and he resuls compared o he analyical soluion. I is shown ha provided sabiliy condiions for boh numerical schemes are no compromised, he Schmid scheme is beer han he Crank-Nicolson scheme a he paricular poin 8% from he conducing end of he rod. Wih he rod discreized ino si poins, boh ends of he rod produce he same resuls for boh numerical schemes. Wih he remaining four poins, i is shown ha hree poins produced values which showed ha he Crank-Nicolson scheme is beer han he Schmid scheme a hose hree poins, bu no he fourh. General Terms Numerical Soluion, Finie Difference, Relaive Absolue Error, Parabolic PDEs, Separaion of Variables, Dirichle condiion, Crank-Nicolson scheme, Schmid scheme, Sabiliy Keywords Numerical Analysis, Algorihm, Hea Equaion. INTRODUCTION Numerical mahemaics has come o he aid of mahemaicians for cenuries and has made he soluion of oherwise unsolvable mahemaical problems quie easy. This has been fel in almos all branches of Science especially, mahemaics, engineering and medicine. When numerical mahemaics reached is peak in he mid h cenury, among hose who made heir names were he German scholar Erhard Schmid (forming he Schmid mehod), English Mahemaical mahemaicians John Crank and Phyllis Nicolson (ogeher forming he Crank-Nicolson mehod). These ousanding mahemaicians of old proposed algorihms for solving parial differenial equaions numerically. Even hough hese algorihms are used widely oday in almos all fields of Science, ime has come o pu hese algorihms under he lens for informed decision o be made on hem. Alhough quie some seps have been aken by various mahemaicians o analyse hese algorihms, all effor unforunaely focus only on heir sabiliy analysis.. OBJECTIVES This paper among oher hings seeks o: Apply analyical and numerical mehods o solve a parabolic parial differenial equaion. Compare he numerical soluions o he analyical soluion and draw informed conclusions abou he numerical algorihms. Compare and conras error analysis and sabiliy analysis.. METHODOLOGY A model problem for an aluminium rod of lengh L, iniially a room emperaure wih one end immersed in boiling waer and he oher insulaed is considered. The subsequen emperaure disribuion across he rod is compued analyically as well as numerically. Numerically, wo algorihms are employed o compare and conras heir performance in erms of efficiency and accuracy: Schmid Scheme Crank-Nicolson Scheme The soluions will be limied by he following condiions so as o achieve uniformiy in soluion. Compuer algebra sysems will be employed o minimize errors if no eradicae i compleely.. JUSTIFICATION Hea equaion has many applicaions in engines and srucural mechanics. I is also used eensively in Biology where i is known as diffusion equaion and models he diffusion of subsances such as drugs, bacerial, or viral spread in he human sysem. This research paper will pu o res he overreliance on sabiliy conceps alone in selecing appropriae numerical algorihms for predicing he behaviour of hea ransfer by he hea equaion. This will also help pharmaciss in predicing he behaviour of cerain drugs in he human body. I will also pu in rerospec which numerical scheme is bes for solving parabolic PDEs in general.. ANALYTI CAL SOLUTION Le be he emperaure in degrees Celsius a a disance from he ho end, minues afer he end a is immersed. Le c be he hermal diffusiviy of he rod. The emperaure is governed by he problem This problem is no correcly formulaed for he separaion of variables echnique because he boundary condiion a

2 Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober is non-homogenous. Observe ha he emperaure profile as is easily deermined by seing equal o and ignoring he iniial condiion. The resul is he seady-sae emperaure problem where is he seady-sae emperaure. The soluion of his simple problem is he consan. Suppose we subrac from o obain a new emperaure variable, say The ime dependen equaion can really be solved a any ime, bu since is ye unknown, le s hold on. Now he spaial problem soluion is There are hree possible scenarios o deal wih here. wih in place of, he problem becomes CASE : In his case he soluion o he differenial equaion is The new problem has homogenous boundary condiions. Simplifying he problem by scaling, a new emperaure and ime is defined by Wih hese changes, he problem becomes Applying he firs boundary condiion gives Applying he second boundary condiion and using he immediae resul yields Going afer non-rivial soluions means To solve he above problem, he firs hing o do is o apply separaion of variables. Firs, assume ha he soluion akes he form Subsiuing Separaing he variables gives The only condiion ha will make he above equaion meaningful is when boh sides evaluae o a consan. ie where is called he separaion consan and is arbirary. The equaion above can be spli i ino he following wo ordinary differenial equaions The ne sep is o make sure ha he produc soluion,, saisfies he boundary condiions so i is plugged ino boh epressions If he firs equaion is considered, i is eiher or. However for every and, ie he rivial soluion, and a soluion o any linear homogeneous equaion, a non-rivial soluion is more desirable. Therefore i is assumed ha infac. Likewise from he second, soluion. In summary o avoid he rivial Noe ha c is no needed in he eigenfuncion as i will ge absorbed ino anoher consan ha will be picked up laer on. CASE : The soluion o he differenial equaion is Applying he boundary condiion gives So in his case he only soluion is he rivial soluion and so is no an eigenvalue for his boundary value problem. CASE : Here he soluion o he differenial equaion is Applying he firs homogenous condiion gives and applying he second gives Assuming and so and his means. Thus, is he only rivial soluion in his case. Therefore here will be no negaive eigenvalues for his boundary value problem. The complee lis of eigenvalues and eigenfuncions for his problem are hen Now solving he ime differenial equaion and noe ha because of simpliciy, is no subsiued. This is a simple linear (and separable for ha maer) s order differenial equaion and he soluion is and noe ha here is no a condiion for he ime differenial equaion and ha is no a problem. Now ha boh he ordinary and parial differenial equaions are solved a final soluion can be wrien. Noe however he

3 Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober infiniely many soluions found since here are infiniely many soluions (ie eigenfuncions) o he spaial problem. The produc soluions are hen The produc soluion is denoed o acknowledge ha each value of n will yield a differen soluion. Also noe ha he h in he soluion o he ime problem is changed o o denoe he fac ha i will probably be differen for each value of n as well and because had he been kep wih he specific eigenfuncion i would have absorbed he c o ge a single consan in he soluion. The principle of Superposiion is no resriced o only wo soluions and so he following is also a soluion o he parial differenial equaion The soluion for he coefficien is given by where. By seing and evaluaing yields Hence Recovering he original soluion and compleely eliminaing L by subsiuion, Eending he soluion furher by aking yields Table. Soluion of he Hea Equaion Using he Analyical mehod

4 Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober 8 Temperaure T (, ) o C Displacemen (f) Figure : A D-Plo of he daa produced in he Analyical soluion. NUMERICAL SOLUTION OF THE HEAT EQUATION To solve he hea equaion numerically, boh he and variables need o be discreized and proceed o deal wih he -variable employing finie difference approimaion. This concep according o hese grea schools of hough Schmid and Crank-Nicolson is presened in his paper.. Schmid Scheme When Dirichle boundary condiions are imposed, hose values mus be specified a he boundary poins. The firsorder forward-ime second-order cenred-space (FTSC) approimaion of he hea equaion is given by Wih he following rigonomeric ideniies Equaion () hen becomes For he original error no o grow, he amplificaion facor is resriced as As sine funcion has he range.. Sabiliy of Schmid Scheme Using Von Neumann/Fourier Mehod Bu ()-() Le Von Neumann considers he homogenous par of he difference equaion This confirms ha provided hen he Schmid scheme will be sable, oherwise i will be unsable. In oher words, he Schmid scheme is condiionally sable. For he purpose of his paper, r is se o. o achieve sabiliy. The able below shows he resuling compuaions using he Schmid scheme. Using Fourier series, he n-componen soluion of he difference scheme is Subsiuing () in () and simplifying yields

5 Table. Soluion of he Hea Equaion using he Schmid Scheme Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober Temperaure T (, ) o C Displacemen (f) Figure : A D-Plo of he daa produced by he Schmid scheme wih r=.. Cranch Nicolson Scheme.. Sabiliy Analysis of he Crank-Nicolson Scheme Using Von Neumann/Fourier Mehod Using Fourier series, he n-componen soluion of he difference scheme is Subsiuing (d) in (c) and simplifying yields Wih he following rigonomeric ideniies Equaion (f) hen becomes Le Guided by he fac ha and

6 Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober This shows ha he Crank-Nicolson scheme is uncondiionally sable bu for he sake of uniformiy and comparison, is chosen. The resul of he Crank- Nicolson scheme compuaion is shown in he able below: Table. Soluion of he Hea Equaion Using he Crank-Nicolson Scheme Temperaure T (, ) o C Displacemen (f) Figure : A D-Plo of he daa produced in he Crank-Nicolson scheme wih r =..

7 Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober 7. DATA ANALYSIS Table. Relaive Errors in Numerical Approimaions Using he Schmid Scheme Temperaure T (, ) o C. Displacemen (f) Figure : A D-Plo of he absolue relaive error wih he Schmid scheme Table. Relaive Errors in Numerical Approimaions Using he Crank-Nicolson Scheme

8 Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober. Temperaure T (, ) o C. Displacemen (f) Figure : A D Plo of he Absolue Relaive Error wih he Crank-Nicolson scheme. 8. DISCUSSION OF RESULTS For an aluminium rod a room emperaure whose one end is insulaed and he oher immersed in boiling waer, one would epec ha wih ime he emperaure will grow uniformly hrough he rod unil such a ime ha he emperaure disribuion is he same across he whole rod. Even hough, he rod was no given enough ime o undergo such a ransformaion, ha is, wih in a ime span of, i was sill evidenly clear ha he emperaure disribuion was acually growing across he rod. For he analyical soluion, he emperaure disribuion a, for, fell below he room emperaure of he rod, which was quie unusual. I begun o improve when, and only a when emperaure acually fell below room emperaure. For and somehing unusual was happening. All he emperaure a ha poin was acually less han he insulaed end of he rod. The aluminium rod begun o ehibi is rue conduciviy characerisics from, when here was ruly uniformiy in hea ransfer. The Schmid scheme conforms o iniial condiions of he rod perfecly. Hea only begins o flow a when he emperaure doubled a he poin while oher pars of he rod remained a room emperaure. The emperaure gradually increases hrough he rod unil when all pars of he rod had eperience emperaure rise. This ime conforms o he ime he rod ehibis is conduciviy properies. The iniial condiions of he rod is perfecly obeyed by he Crank-Nicolson mehod wih he ecepion ha. This defeas he fac ha. This iniial condiion is however saisfied from whiles eperiences a decrease in emperaure. This negaive phenomenon decreases in he range. The conduciviy propery of he rod hen sars from. Generally, he values obained wih he Schmid and he Crank-Nicolson schemes compare favourably wih values obained wih he analyical mehod of soluion. The rue value for he final soluion for he Analyical, Schmid, and Crank-Nicolson schemes is shown in Table ; 8

9 Inernaional Journal of Compuer Applicaions ( ) Volume 79 No, Ocober Table. True Values of he Various Mehods o he Hea Equaio Analyical Scheme Schmid Scheme Crank-Nicolson Scheme The above able clearly shows how close he numerical soluions wih boh he Schmid scheme and he Crank- Nicolson scheme are o he Analyical soluion. Shear observaion of he above able shows which numerical scheme is closer o he analyical scheme. Bu he bes guide is he resul from he absolue relaive errors of he wo numerical schemes, which is shown in able 7. Table 7. Absolue Relaive Error for he Final Soluion of he Hea Equaion. Schmid Scheme Crank-Nicolson Scheme Boh schemes absolue relaive error a he boundary poins, ha is, and are he same. In he range, he Crank-Nicolson scheme is he bes represenaion of he Analyical soluion. However for, he Schmid scheme seems o perform beer han he Crank-Nicolson scheme. However on a scale of, he Crank-Nicolson scheme will occupy 7 while he Schmid scheme will occupy. This alone canno however inform he choice of one over he oher because if one considers a paricular poin alone on he rod, he Schmid may be beer han he Crank-Nicolson scheme. 9. CONCLUSIONS A all discree poins of he rod a he given ime frame, he analyical soluion fails o conform o he iniial condiions of he problem. This is so because he emperaure of he rod a he insulaed end a ime is supposed o be whiles he remaining poins were supposed o be very close o he iniial emperaure wih he ecepion of he end immersed in boiling waer. Ironically, he emperaure a poin was somehow closer o he room emperaure han a he poin. This is surprising since hea ransferred from he immersed end of he rod acually reduces as i ges o he insulaed end. The analyical mehod however suggess ha wihin he range emperaure was rising insead of falling. There was however ecepion for he ime frame a poins on he rod where he behaviour conducing propery of he rod was obeyed. The Schmid scheme obeys and conforms perfecly wih he iniial condiions of he rod, mainaining he iniial emperaure of a all poins of he rod wih he ecepion of he immersed end which assumes he emperaure of he boiling waer. This presens he Schmid scheme as a huge improvemen of he analyical mehod. The Crank-Nicolson scheme complies well wih he iniial condiions ecep a he boundary when i is supposed o be bu he numerical soluion acually gives. Per he Analyical mehod, he rod s conduciviy drops in he range for he ime before i rises again. For he Crank-Nicolson scheme, he anomaly in hea conducion is seen a in he ime inerval where as he Schmid scheme had no anomaly. Wih r specially chosen such ha he sabiliy of boh he Schmid and Crank-Nicolson schemes is no compromised, he claim of superioriy of one scheme over he oher is far feched. Infac superioriy can only be claimed wih respec o a paricular discree poin in quesion. The Schmid scheme has shown o be more reliable han he Crank-Nicolson scheme which is srangely seen as an improvemen of he Schmid scheme.. REFERENCES [] Glenn Ledder.. Differenial Equaions: A Modeling Approach. McGraw-Hill. [] John H. Mahews and Kuris D. Fink. Numerical Mehods Using MATLAB. Pearson Educaion, Inc. [] Paul Dawkins. 7. Lecure Noes on Differenial Equaions. hp://uorial.mah.lamar.edu. [] David Richards and Adam Abrahamsen.. The One Dimensional Hea Equaion. [] Alfio Quareroni, Riccardo Sacco and Fauso Saleri.. Numerical Mahemaics. Springer-Verlag New York, Inc. [] Golub, Gene H., and James M. Orega. 99. Scienific Compuing and Differenial Equaion: An Inroducion o Numerical Mehods. San Diego: Academic Press. IJCA TM : 9