zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Hopscotch method: The numerical solution of the Frank-Kamenetskii partial differential equation. (English) Zbl 1205.80086
Summary: Numerical solutions to the Frank-Kamenetskii partial differential equation modelling a thermal explosion in a cylindrical vessel are obtained using the hopscotch scheme. We observe that a nonlinear source term in the equation leads to numerical difficulty and hence adjust the scheme to accommodate such a term. Numerical solutions obtained via MATLAB, MATHEMATICA and the Crank-Nicolson implicit scheme are employed as a means of comparison. To gain insight into the accuracy of the hopscotch scheme the solution is compared to a power series solution obtained via the Lie group method. The numerical solution is also observed to converge to a well-known steady state solution. A linear stability analysis is performed to validate the stability of the results obtained.
MSC:
80M20Finite difference methods (thermodynamics)
80A20Heat and mass transfer, heat flow
35Q79PDEs in connection with classical thermodynamics and heat transfer
65M12Stability and convergence of numerical methods (IVP of PDE)
Software:
DIMSYM; Matlab
References:
[1]Anderson, D.; Hamnén, H.; Lisak, M.; Elevant, T.; Persson, H.: Transition to thermonuclear burn in fusion plasmas, Plasma phys. Contr. fusion 33, No. 10, 1145-1159 (1991)
[2]Baumann, G.: Symmetry analysis of differential equations with Mathematica, (2000)
[3]Bluman, G. W.; Kumei, S.: Symmetries and differential equations, (1989)
[4]Bluman, G. W.; Cole, J. D.: J. math. Mech., J. math. Mech. 18, 1025 (1969)
[5]Britz, D.; Baronas, R.; Gaidamauskait&edot, E.; ; Ivanauskas, F.: Further comparisons of finite difference schemes for computational modelling of biosensors, Nonlinear anal.: model. Contr. 14, No. 4, 419-433 (2009)
[6]Chambré, P. L.: On the solution of the Poisson – Boltzmann equation with application to the theory of thermal explosions, J. chem. Phys. 20, 1795-1797 (1952)
[7]Crank, J.; Nicolson, E.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. camb. Phil. soc. 43, 50-67 (1947) · Zbl 0029.05901
[8]Crank, J.; Furzeland, R. M.: The treatment of boundary singularities in axially symmetric problems containing discs, J. inst. Math. appl. 20, No. 3, 355-370 (1977) · Zbl 0367.65050 · doi:10.1093/imamat/20.3.355
[9]Dresner, L.: Phase-plane analysis of nonlinear, second-order, ordinary differential equations, J. math. Phys. 12, 1339-1348 (1971) · Zbl 0266.34034 · doi:10.1063/1.1665739
[10]Evans, D. J.; Danaee, A.: A new group hopscotch method for the numerical solution of partial differential equations, SIAM J. Numer. anal. 19, No. 3 (1982) · Zbl 0486.65061 · doi:10.1137/0719039
[11]Feldberg, S. W.: J. electroanal. Chem., J. electroanal. Chem. 222, 101 (1987)
[12]Frank-Kamenetskii, D. A.: Diffusion and heat transfer in chemical kinetics, (1969)
[13]Gaidamauskait&edot, E.; ; Baronas, R.: A comparison of finite difference schemes for computational modelling of biosensors, Nonlinear anal.: model. Contr. 12, No. 3, 359-369 (2007)
[14]Gordon, P.: Nonsymmetric difference equations, SIAM J. Appl. math. 13, 667-673 (1965) · Zbl 0144.18104 · doi:10.1137/0113044
[15]Gourlay, A. R.: Hopscotch: a fast second order partial differential equation solver, J. inst. Math. appl. 6, 375-390 (1970) · Zbl 0218.65029 · doi:10.1093/imamat/6.4.375
[16]Gourlay, A. R.: Some recent methods for the numerical solution of time-dependent partial differential equations, Proc. royal soc. London A 323, 219-235 (1971) · Zbl 0234.65080 · doi:10.1098/rspa.1971.0099
[17]Gourlay, A. R.: Splitting methods for time dependent partial differential equations, The state of the art in numerical analysis, 757-791 (1977)
[18]Gourlay, A. R.; Morris, J. Li.: Hopscotch difference methods for nonlinear hyperbolic systems, IBM J. Res. develop. 16, 349-353 (1972) · Zbl 0268.65056 · doi:10.1147/rd.164.0349
[19]Gourlay, A. R.: Recent developments of the hopscotch method, Advances in computer methods for partial differential equations II, 1-6 (1977)
[20]Harley, C.; Momoniat, E.: Steady state solutions for a thermal explosion in a cylindrical vessel, Mod. phys. Lett. B (MPLB) 21, No. 14, 831-841 (2007) · Zbl 1115.80005 · doi:10.1142/S0217984907013250
[21]Harley, C.; Momoniat, E.: Instability of invariant boundary conditions of a generalized Lane – Emden equation of the second-kind, Appl. math. Comput. 198, 621-633 (2008) · Zbl 1146.34031 · doi:10.1016/j.amc.2007.08.077
[22]Head, A. K.: Comput. phys. Commun., Comput. phys. Commun. 77, 241 (1993)
[23]Hundsdorfer, W. H.; Verwer, J. G.: Linear stability of the hopscotch scheme, Appl. numer. Math. 5, 423-433 (1989) · Zbl 0684.65089 · doi:10.1016/0168-9274(89)90041-X
[24]Ibragimov, N. H.: CRC handbook of Lie group analysis of differential equations, CRC handbook of Lie group analysis of differential equations 1 (1994)
[25]Mikhail, M. N.: On the validity and stability of the method of lines for the solution of partial differential equations, Appl. math. Comput. 22, 89-98 (1987) · Zbl 0619.65105 · doi:10.1016/0096-3003(87)90038-5
[26]Momoniat, E.; Mcintyre, R.; Ravindran, R.: Numerical inversion of a Laplace transform solution of a diffusion equation with a mixed derivative term, Appl. math. Comput. 209, 222-229 (2009) · Zbl 1161.65351 · doi:10.1016/j.amc.2008.12.037
[27]Momoniat, E.: A thermal explosion in a cylindrical vessel: a non-classical symmetry approach, Int. J. Mod. phys. B 23, No. 14, 3089-3099 (2009) · Zbl 1170.80332 · doi:10.1142/S0217979209052790
[28]G.R. McGuire, M.Sc. Thesis, University of Dundee, 1970.
[29]Mcguire, G. R.; Morris, J. Li.: Explicit – implicit schemes for the numerical solution of nonlinear hyperbolic systems, Math. comput. 29, No. 130, 407-424 (1975) · Zbl 0312.65062 · doi:10.2307/2005560
[30]Ovsiannikov, L. V.: Group analysis of differential equations, (1982) · Zbl 0485.58002
[31]Rice, O. K.: The role of heat conduction in thermal gaseous explosions, J. chem. Phys. 8, 727-733 (1940)
[32]Rice, O. K.; Campbell, H. C.: The explosion of ethyl azide in the presence of diethyl ether, J. chem. Phys. 7, 700-709 (1939)
[33]Steggerda, J. J.: Thermal stability: an extension of Frank-kamenetskii’s theory, J. chem. Phys. 43, 4446-4448 (1965)
[34]Sherring, J.; Head, A. K.; Prince, G. E.: Dimsym and Lie: symmetry determining packages, Math. comput. Mod. 25, 153-164 (1997) · Zbl 0918.34007 · doi:10.1016/S0895-7177(97)00066-6
[35]G.L.G. Sleijpen, Strong Stability Results for the Hopscotch Method with Applications to Bending Beam Equations, Utrecht. · Zbl 0663.65091 · doi:10.1007/BF02259092
[36]Taha, T. R.; Ablowitz, M. J.: Analytical and numerical aspects of certain nonlinear evolution equations II, Numer. nonlinear Schrödinger equat. Reprinted J. Comput. phys. 55 (1984) · Zbl 0541.65083 · doi:10.1016/0021-9991(84)90004-4
[37]Zeldovich, Y. B.; Barenblatt, G. I.; Librovich, V. B.; Makhviladze, G. M.: The mathematical theory of combustion and explosions, (1985)