The Goursat problem, associated with hyperbolic partial differential equations, arises in several areas of applications. Several finite difference schemes have been proposed to solve the Goursat problem

Md. Ismail, Ahmad Izani

Nasir, Mohd. Agos Salim

Talib, Abdullah Zawawi

English

The Goursat problem, associated with hyperbolic partial

differential equations, arises in several areas of

applications

Md Ismail, Ahmad Izani

Nasir, Mohd Agos Salim

Repository@USM

ppceedlngs of thc IASTED Internrtlond ConfcrrnceAPPLIED SIMULATION AIID MODELLINGJune 2&30' 20Of' Rhodcs, GrceceAbstractThe Goursat problern, associated with hyperbolic partialdifferential equations, arises in several areas ofapplications. Several finite difference schemes have beenproposed to solve the Goursat problem. Amongst theseschemes is a scheme which implements harmonic meanaveraging of function values. A comparative study whichhas been carried out concluded that hafmonic meanaveraging yielded more accurate rcsults than arithmeticmean averaging. However, there seemed to bediscrepancies between the conclusions and the displayedresults. In this paper, we present the results of acomparative study which we have conducted on threeGoursat problems over a range of grid sizes. Our resultsindicate that arithmetic mean averaging is more accuratethan harmonic mean averaging. We also show thatarithmetic mean averaging has an advantage when appliedto linear Goursat problems.Key WordsGoursat problern, finite difference schemes. arithmeticmean, harmonic mean1. IntroductionMany mathematical models in science and engineeringnecessitate the solution of a partial differential equation.As analytical solutions are often difficult to obtain,numerical methods (such as the finite difference or finiteelement method) are frequently used.The Goursat problern, associated with hyperbolic partialdifferential equations, arises in several aieas of p-hysicsand engineering. Frisch and Chao [l], Cheung [2], Kaup3id Newell t3l, An and Hua t4l, Hillion [5i,McClaughlin et. al. [6], Chen and ii 121, Kaup indSteudel [8] describe in detail areas of appliiaiions where aGounat problem arises. Several finite difference schemeshave been proposed to solve the Goursat problem.Amongst these schemes is a scheme which implementsharmonic mean averaging of function values. Acomparative study which has been carried out t9]concluded that this approach yielded more accurateresults compared with the use of the standard method ofanthmetic mean averaging.443_056 243In this pap€r we study the accuracy of a finite differenceschemc based on harmonic mean averaging and a finitedifferencc schemc based on arithmetic mean averagingwhen applied to threc (one linear, one nonlinear aad onewith a derivative tcrm) Goursat problems.2. The Goursat Problem and FiniteDifference SchemesThe Goursat problem is of the form [9]:tury = .f (x,y,tt ur,uy)z(x,0) = $(x), z(0 , y) = ty(y), fiO) = ry(0) ( l)0(x( a,0<xSbThe established finite difference scheme is based onarithmetic mean (AM) averaging of function values and isgiven by (W amraz, 1993):Ui+l.i+l *Ui,i -Ui+l,i -ut,t*,h2 e)= Lo(!,u.,u + fi.i + fi,r.i * -f,.iu)h denotes the grid size. Henceforth, we shall rcfer to thefrnite difference scheme (2) as the AM scheme. For theAM scheme, the function value at gdd location(i +l / 2, j +l / 2) is given by:tl - \-o\f,t.i-t + -fi.i + -f,q,i + -ft.1t) (3)Wazwaz (1993) presented a new scheme for the Goursatproblem. This scheme is based on harmonic meanaveraging of function values and is given by:ui*t.i't * ui.i -ui,t.i -ui.i-th24.f ,t.1t -f ,.1f ,.t-, f ,.r. if , -r. i t.f ,.i (f ,q. i + f i. i *r\ + -f it. i.f ,. i t ( -f i t. i -r + f i. i ))h+tNUMERICAL SOLUTION OF TIIE GOT]RSAT PROBLEMAbmad k+ Mq. Ismailr, U9n!.-fS9s SalimNasirr, AMultah Zawawi Talib2'School of Mathematical Sciences. 2 School ofCornputer SciencesUniversiti Sains MalaysiaI1800 USM Penang, Malaysia(4)Henceforth, we shall refer to the finite difference scheme(4) as the HM scheme. The harmonic mean (HM) of anytwo real numbers a and b i, 20b . The function valuea+bat location (t+tt2,j+ll2),i.e. the r.h.s of equation(4). is obtained from;HM(HM of .f,.j*, andf ,-,., ;HM of .f,_r.1*rand.f,., 1(s)Wazwaz [9] stated that he investigated the application ofthe AM and HM scheme over a wide range of examplesand concluded that the HM scheme appears to give befter:sults (in terms of accuracy). However, results were only- resented for the non-linear Goursat problem (withh=0.05);u\'= et'ru(x,0) =:- ln(l + e.').,u(0,,')=:-ln(l+s''';L0<x<4,0<y<4The results presented were the relative errors at 16selected grid points. However, from an examination of thedisplayed results we observed that the AM scheme wasmore accurate than the HM scheme for all 16 grid points.The question that arises is whether the conclusion that theHM scheme was more accurate was based onerformance at grid points not displayed (for h=0.05,..rere are 6400 grid points). We aim to compare theaccuracy of the AM and HM schemes, for a particulargrid size. by computing the number of grid points arwhich one scheme is more accurate than the other and thcaverage relative error over all -erid points. This l.ill bcimplemented over a range of grid sizes on problem (6)a.well as the linear Goursat problem:ll\. =lltl(.r,0) = €''ra(0,.t,) = gl0<x<2,0<(-landurr=-l+y+uu (-r,0) = -1 + e'tr (0,,tr) = -l + e'0<x<2.4.0>v<2.4In this way we hope to draw firmer conclusions regardingthe accuracy of the AM and HM schemes. Analyicalsolutions for (6), (7) and (8) can be found in Wazu,azt101.3. Numerical ExperimentsComputer programs for problems (6), (7) and (8) weredeveloped. For the non-linear Goursat problem (6) with h:0.05. we obtained:Table l: Relative errors for the AM scheme. h:0.05x 1.0 2.0 3.0 4.01.02.03.04.07.28907 I 3e-0059.7076691e-0056.4976523e-0054.038567 le-0059.7076691e-0052.9467327e-0043.4128209e-0042.5890833e-0046.4976523e-0053.4128209e-0048.0424181e-0049.5132232e-0044.038567 le-0052.5890833e-0049.5132232e-OO42.1149666e-003244Table 2: Relative errors for the HM scheme, h=0'059.0306034e-005l.8l 10830e-0042.008 I I 84e-0041.757065 le-0041.8 I 10830e-0045.3529258e-0047.95 l9 l42e-0047.6 I 82887e-0042.0081 l84e-0O47.9519142e-0ML9t928l0e-0032.5539158e-0031.757065 le-0047.6182887e-0042.5539158e-0035.7780549e-003We also computed that the:Number of grid points where the AM scheme is superior= 6400Number of grid points where the HM scheme is superior=QAverage relative error of the AM scheme = 2.5268253e'004Average relative error of the HM scheme = 6.2442268e'004For grid sizes h:0.025,0.1, we obtained the followingresults:Table 3: Results for Problem (6) with h= 0.025, 0.1For the linear Goursattbllorving results:problem (7), we obtained theTable 4: Results for Problem (7) with h=0'025' 0'05, 0'lFor the Goursat problem (8), we obtained the followingresults :Table 5 : Results for Problem (8) with h = 0'003' 0'006'0.03From the above results (and the results for h values notdisplayed in this paper) it is clear that the AM scheme ismore accurate than the HM scheme'h = 0.025 h = 0.05 h = 0.1No. of gridpoints forwhich theAM schemcis suoerior6400 1600 400No. of gridpoints forwhich thcHM schcmcis supcrior0 0 0Avcragerclativc enorof the AMscheme4.3506679e405 t.7771664e-004 7.4t07470f-0MAveragerclative errorof the HMscheme8.6977635e-005 3.5484552e-0O4 1.472382 I e-003h = 0.025 h = 0.1\o. of grid pointstbr which AMscheme is suDerior15600 I 600No. of grid pointslor which HMscheme is suDenor0Al'erage relativeernrr of the AMschcnle6.11987 I 8e-005 1.0187387e-001Arerage relativeerror of the HMsc nente1.5177398e-004 2.5(r987 7e-00-lh = 0.003 h = 0.006 h = 0.03No. of gridpoints forwhich theAM schemeis superior6t7647 l 58706 6r90No. of gridpoints forwhich theHM schemeis suoenor23s3 t294 210Averagerelativeerror of theAM schemeI .002.17 l4€-l 2.0028735ex-3 9.934 I 687ex-lAveragerelatrveenor of theHM scheme5. I 36Ls57ex-2 5.l9.16ll7ex-2 5.6393552ex-22454. Implementation AspectsConsider the linear Goursat problem (7) as an example. Ifthe AM scheme is used. we obtain the finite differencescheme:u i_.t. 1_t + u,. 1 - u i*1. 1 - u i. j-tEquation (7) is a linear equation which can easily besolved for the unknown u i*1. i.rthe harmonic mean scheme is used we obtain:u i\. i*t * u i. i - u i-t. i - ui. j*tll4u i*1.1,]u ;.,u i. i+'ru i+r. ju i *t, i *tu i..1 (u i +r. 1 + u i.i +r ) * u i*t. ju i. j *t (Li*t.7*t + u i. j )(10)This is a non-linear equation in the unknown Ll,r,.r*, andwould require iteration, with its associated computationalcosts, for its solution. We thus see that the AM schemehas an advantage in that it preserves the linearity of alinear Goursat problem and consequently thestraightforward solution procedure.i. ConclusionsIn this paper we have studied the AM and HM finitedifference schemes for the solution of the Goursatproblem. A previous comparative study concluded thatthe HM scheme was more accurate. However, thedisplayed results indicated otherwise. Our investigations.involving the computation of the number of points atwhich one scheme was more accurate than the other andthe comparison of the average relative error for threeGoursat problems, have found that the AM scheme ismore accurate. We further make the observation that forlinear problems the AM scheme has an advantage in thatit preserves the linearity of linear Goursat problems.AcknowledgementsWe acknowledge the financial support of an FRGSresearch srant.Referencestll R.A. Frish and B.R. Chao. On a bounded one-dimensional Poisson-Vlasov System. S7-l 1/Journal of Applied Mathentatics. 1973.T.Y. Cheung. Three non-linear ivp's of thehyperbolic type.SL4M Journttl qf NumeritulAnalysis.I978.t3l D.J. Kaup & A.C. Newell. The Goursat andCauchy problems for the Sine Gordon Equation..SIAM Jcsurnal o/ Applied Mathematics, 1978.Y.L-An. W.C.-Hua. The discontinuous ivp of areacting gas flow system.. 7'ransactions o/' rhe.Anteric'an Muthemqtit'al Soc'ietr'. 198 l.P. Hillion. A note on the derivation of paraxialequation in nonhomogeneous media, SIAM Journolof .4pplied Mathematics. 1992.J.R. Mclaughlin, P.L. Polyakov, P.E. Sacks.Reconstruction of a spherical symmetric speed ofsound, SIAM Journal of Applied Mathematics.t994.S.Chen and D.Li, Supersonic flow past asymmetrically curved cone, [ndiana Universitt'Ma themat ic s Journa l, 2000.D.J. Kaup and H.Steudel, Virn:al Solutions and theasymptotics of second harmonic generation.Inverse Problems, vol I7,2001 .A.M. Wazwaz. On the Numerical solution for theGoursat Problems, Applied Mathematics andComputation, 1993.A.M. Wazwaz, The decomposition method forapproximate solution of the Goursat Problem.Applied Mathematics and Computation, 1995ll0lt)lt-lh) (9)tl \= -\u,.t. i.t I u i. i + u i.l, i + u i. i-l )t4ltslt6l17lt8ltel246A Publicotion of The Inlernotionsl Assoriotion ofScience qnd Technology {or Development - IASTEDTABLE OF CONTBNTSASM 2OO4PowERANo nNnncY SYSTEMS I443-002: A Thermodynamics Method of OnlineCalculation Steam Turbine Exhaust Enthalpy withlmproved Fltigel FormulaZ. Han and K Yang ........... .........................1443-023 The Application of the Diesel Engine Simulatorfor the Diagnostic TrainingS. Kuj ...............7443-157: Control of Combustion based onNeuro-fuzzy ModelZ. H{mer, G. Devinyi, J. Kovdcs, and U. Kortela ..........13443-16l: PEM Fuel Cell Model for SystemSimulation and OptimisationS.S.H.U. Gamage, P.R. Palmer, and B. Lakeman..........l8443-095: A Model Regarding Electrical Contactsin Advanced DegradationC.G. Aronis, C.P. Psomopoulos,C.G. Karagiannopoulos, and P. D. 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A note on the derivation of paraxial equation in nonhomogeneous media,

On a bounded onedimensional Poisson-Vlasov System. S7-l 1/ Journal of Applied Mathentatics.

On the Numerical solution for the Goursat Problems, Applied Mathematics and Computation,

Supersonic flow past a symmetrically curved cone, [ndiana Universitt' Ma themat ic s Journa l,

Y.L-An. W.C.-Hua. The discontinuous ivp of a reacting gas flow system.. 7'ransactions o/' rhe .Anteric'an Muthemqtit'al Soc'ietr'.

Numerical Solution Of The Goursat Problem.

616 WSEA3 TRANSACTIONS ON MATHEMATICS Issue 3, Vol. 3, July 2004 ISSN: 1109·2769MORn: AGOS SALIM NASIR}, AHMAD IZANI MD. ISMAIL2IDepartment ofEngineering Sciences,Universiti Teknologi MARA,Penang Branch Campus, 13500 Pennatang Pauh, Pulau Pinang,·MALAYSIA2School ofMathematicai Sciences,Universiti Sains Malaysi~ 11800 Minden, Pulau Pinang,MALAYSIA.mohdagossalimnasir@yahoo.com, izani@cs.usm.myAbstract:-The Goursat problem, associated with hyperbolic partial differential equations, arises in several areas..a( applications. These include mathematical modeling of reacting gas flows and supersonic flow. Recent~erical studies of the problem have focused on the implementation and accuracy aspects of various finitedifference· schemes. ~eoretical considerations such as· ~abi1ity, consistency and convergence have notreceived much attention. In this paper we consider the theoretical aspects.ofa numerical scheme widely used tosolve the problem by considering its application to model linear problem . We obtain results relating to thestability (using Von Neumann stability analysis), consistency and convergence of the scheme. We verify thesetheoretical results with data from computational experiments.Key- Words:- Finite difference schemes; partial differential equations; means; stability;Consistency; convergence1 IntroductionThe Goursat problem, associated with hyperbolicpartial differential equations, arises· in several areas ofphysics and engineering. An and !Ida (1981) and Chenand Li (2000) describes in detail how this equationarises and is used in the mathematical modeling of&ting gas flows and supersonic flows respectively.~ investigate these mathematical· models in greaterdetail, we need to resort to numerical techniques and inthis regard the finite difference method is often used.The standard· finite difference scheme for the Goursatproblem is a method .based on arithmetic averaging offunction values and a recent study (Ismail et. aI., 2004)has reaffinned the advantages of this method (" theAM scheme").What a~surance is there that the solution obtained by anumerical method is close to the exact solution? Oneway would be to check that the computed resultsconverge when the grid sizes are reduced. However..clearly it would .be better if convergence can beguaranteed beforehand. It is well-lmown in numericalanalysis that for linear .problems this guarantee can begiven if the numerical method is stable and consistent.The concept of stability is concerned with the growth, ordecay, of errors (produced because the computer cannotgive answers to an infinite number of decimal places) atany· stage of the computation (Fletcher, 1990). Thesystem of algebraic equations generated by thediscretisation process is said to be consistent with theoriginal partial differential equation if, in the limit that thegrid spacing tends' to zero, the system of algebraicequations is equivalent to the partial differential equarior.at each grid point (Fletcher, 1990). Consistency isconcerned with how well the algebraic equationsapproximates the partial differential equation.Recent studies of numerical methods for the Goursarproblem ( Ismail et.a!., 2004 ; Wazwaz, 1993 ) ha\'eWSEAS TRANSACTIONS ON MATHEMATICS Issue 3 Vol. 3, July 2004 JSSNj 1109·2769 617focused on implementation and accuracy of varioustinite' difference schemes. In this paper we willUlvestigate the stability, consistency and convergenceof. the AM' scheme when applied to a model linearGoursat problem. .., The Goursat Problem and the.-AM SchemeI The Goursat problem is of the form rv.razwaz, 199~):U lty =f(x,y,u,u,pu y )u(x,O) = cp(x), u(O, y) =\V(y), 4>(0) =weO)°S x S a,O ~ y S b... (1)The established finite difference scheme is based onI arithmetic mean averaging of function values and isgiven by (Ismail et.a12004):±(fi+1,j+l + fi,j + fi+1,j + fi,i... l ) ..·... (2). j denotes the grid size. As was mentioned earlier, weshall refer to the finite difference scheme (2) as the M1scheme. For the AM scheme, .the function value at 'grid:ocation (i + 1/ 2, j +1/ 2) is given by :The analytical solution of (4) is e HY (Wazwaz, 1995)the AM scheme for the partial differential equa~on in (4is:'U i+1,j+J +,Ui,j - Ui+l,j - Ui,j+-l _h2 -±(~i+I.;'1 +ui.J +Ui+I.J +Ui•J+1)...(5JEquation (5) can be rewritten as:...(6)l+r ' h 2where A=-- with r=->O.1-r 4~ StabilityThe stability of a fmite difference scheme can beinvestigated using the Von Neumann method (Fletcher,1990). In this method, the errors distributed along gridlines at one time level are expanded as a finite Fourierseries. If the separate Fourier components of the errordistribution amplify in progressing to the next time level,then the sch~me is unstable.The error equation for equation (6) is:where <;i,i is the error at the (iJ) grid point. We writeC; i,j as }} e.He.i where A. is the "amplification factorfor" the rnth Fourier mode of the error distribution as it·propagates one step forward in time and em = m1th. Forlinear schemes it is. sufficient to consider the propagationof the error due to just a single term of the Fourier seriesrepresentation i.e. the subscript m can be dropped. ""... (3)-\\"e shall investigate the stability consistency and~onvergence of the A...\1 scheme for linear Goursat~roblems by considering the model linear Goursat~roblem: "Ux.y =uC;i...l.j...l =A(Si+l,j + C;i,j+r) - <;ij ...(7)u(x.O) =e''(u(O. y) =e Yo::; x S 2. 0 ~ y::; 2... (4)Substituting <;i.j =')e.j:j8i into equation (7) gives;618 WSEAS TR.A.NSACTJONS ON MATHEMATlCS Issue 3, Vol. 3 July 2004 ISSN: 1109·2769.(9)Substituting the exact solution into (5) leads to:[note:.- all terms involving u in equation (10) areevaluated at (Xi' Yj)]5 Convergence.- .(101.As h~ 0, equation (10) becomes U x..j = u. Thus _thecondition for consistency is satisfied.A solutio~ of the algebraic equations which approximate. a' given partial differential equation is said to be.convergent if the approximate solution approaches theexact solution for each value of the independent variablesas the .grid spacing tends to zero. The Lax EquivalenceTheorem- (Richtmyer and Morton, 1967) states that given. a· properly (well) posed linear initial value problem and afinite :difference equation that satisfies the -consistencycondition, stability is the necessary and sufficientcondition for convergence. .required thatwhich impliesAe..H9 -1A=~=--­e.r::ie - A...(8)"J1-~ e ..[.Tl)(iTl) = A ())e.r::i 9(iTl) + A(jTl) e .r::iei ). ..,) e.J:i 6ii.e.4 ConsistencyTo test for consistency) the exact solution of the parti~ldifferential equation is substituted into the fmiteOfference scheme and values at grid points· expan~edas a Taylor· series. For consistency, the exp~esslOnobtained should tend to the partial differential equationas the grid sIze tends to zero (Twizell, 1984)..Squaring both sides and after some manipulation weobtain that for IAI ~ 1va) A must'satisfy A '1. ;;:; 1 . Since.•= 1+ r and r > '0) we obtain that the scheme isl-rstable Vr (except r =1)For stability . it .isq Ae.f=ie -1AI ~ I'Ve i.e../=iI ::;1"1'6e -19 -A.Ie.r::ie - AI ~ IAeJ=ie ~ 1\Expanding as a Taylor series about (xj'Yj) gives: That the Goursat problem -is wen posed was establishedby Garabedian (1964)' by transforrrung it into arintegrodifferential .equation and then solving by .. theteclmique of successive approximations. We: haveestablished that the AM scheme for the linear Goursat. problem (4) is both stable and consistent. Thus fro~ t~~·Lax .Equivalence Theorem we can conclude It l~convergent.WSEAS .TRANSACTIONS ON MATHEMATICS Issue 3. Vol. ). July 2004 ISSN: 1109·2769 6196 Computational ExperimentsA computer program using the AM scheme to solve (4)was developed and the computed results are as follows:- absolute errors ·at grid pointsh (0.25.0.25) (0.5.0.5) (0.7S.0.7S) ..{I.0.1.0)0.500 6.923387Se- 6.9233875e- 2.0437S96e- 2.0437S96e-003 003 002 0020.150 S.lSS7520e- 1.7070573e- 3.24933S4e- 4.9962137e-004 003 003 0030.\00 5.5301 S66e- 2.7212902e- S.7076620e- 7.9460331 e-005 004 004 004- v.OSO 2.068406Se- 6.799709ge- 1.2925661 c- 1.9848210e-OOS 005 ·004 004·-0.025 5.17056Se· 1.6997083e- 3.230.&63Se- 4.9610004e-006 005 005 005-.abs·olute errors at grid pointsh (1.25.1.25) ·(L5~l.S) (1.75,1.75) (2.0,2.0)0.500 3.6461034e- 3.6461304e- S.3882494e- 5.3882494c-~ 002 002 002 . 002·'0.150 6.8729164e- 8.8393041 e- 1.087220ge- 1.2957280e-003 003 002 0020.100 1.1530440e- 1.402621ge- 1.7883943e- . 2.0515447e-003 003 003 0030.050 . 2.7267983e- 3.502.446ge- 4.3025 I04e- 5.121255ge-004 . 004 004 0040.025 6.8 152731e- 8.7535555e- 1.0752707e- 1.2798397e-005 OOS 004 004Table 1: h values and absolute errors at various gridpoints.Although tesults at only ,eight points are displayed,numerical experiments indicate that the absolute errorbecomes smaller as· his decreased for all grid pointstested. '7 Conclusionsaevious studies of the finite difference solution of the,"oursat problem· have focused on the accuracy and:mplementation aspects. In'this paper we have studiedthe theoretkal aspects of the finite difference solution'.of a linear Goursat problem .using the Mf scheme.l'sing the Von Neumann method we have sho\\TI that itIS unconditionally stable and we have also sho'Wn it isconsistent. Invoking the Lax Equivalen.ce Theorem wededuce that· the scheme is convergent. Numericalexperiments that we have conducted verify that thesd'.eme is convergentAcknowledgementsWe acknowledge ~e support of an FRGS grant.References1. Y.L-An,W.C.-Hua, The discontinuous ivp of freacting gas flow system, Transactions of theAmerican M~thematicalSociety, 1981.2. S.Chen· and D.Li. Supersonic flow past asyInmetrically curved cone, Indiana UniversityMathematics Journal, 2000.3. C~A.l~ Fletcher, Computational Techniques forFluid· DYnamics, Vol.' 1 2nd ed., Springer Verlag,1990.4. P. 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Gonzalezl'tlathematical Characteristics for the Automated Recognition of ~fusiC31Recordings 698G. ROUSSOpOlilos, D. Fragoulis, C, Papaodysseus, Atlr. Panagopoulos, M. ExarhosThe Best Values of Parameters in Accelerated Successive Overrelaxation Methods 705_l\1olzammad Taghi Dan-ish;On the ~linimalRank Completion Problem for Pattern ~1atrices 711Rafael Canto, :\-facatena Boix, Beatriz RicarteDefiniteness Conditions in the l\lultiparameter Spectral Theory 717Alian Asianol'Generalized Inverses and Color Theory. 721Dirk HuylebroLtckSelf-Similarity Analysis of the Fields Diffracted from Zone Plates with Complex Geometry 726, Diana Calloa Jfendez..Hano LehmanA graph based Clustering method using a hybrid evolutionary algorithm' 731Selle Glinduz OgzidllCtt. A. Sima [JyarIII III, ~SS" 1199-2769

Computational Methods for Partial Differential Equations, Elfis Horwood,

Di.fference ,Methods' for Initial Value Problems, Wiley Interscience,

Kantoroyich Type Generalization Cor Certain ClassorPositive IJnearOperators 007 Alxh,llahAltin,

Onthe Numerical solution for the Goursat Problems, Applied Mathematics and Computation,

Partial Differential Equations,

S.Chen· and D.Li. Supersonic flow past a syInmetrically curved cone,

The discontinuous ivp of f reacting gas flow system,

http://eprints.usm.my/8614/1/Numerical_Solution_of_a_Linear_Goursat_Problem_Stability_Consistency_and_Convergence_(PPSM).pdf

Numerical Solution Of The Goursat Problem.

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