×

Quadratic spline collocation for one-dimensional linear parabolic partial differential equations. (English) Zbl 1189.65235

The authors focus on numerical methods for general linear differential equations of parabolic type posed on one space dimension. The numerical methods rely on quadratic-spline collocation combined with the classical finite difference method. The main results of the paper provide stability and convergence properties of the discretized solution. The final part of the article contains various numerical experiments that sustain the theoretical findings. The new methods developed by the authors are also applied to American put option pricing problem in the paper with satisfactory results.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
91G60 Numerical methods (including Monte Carlo methods)

Software:

BACOL
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Archer, D.: An O(h 4) cubic spline collocation method for quasilinear parabolic equations. SIAM J. Numer. Anal. 14, 620–637 (1977) · Zbl 0366.65054 · doi:10.1137/0714042
[2] Bialecki, B., Fairweather, G.: Orthogonal spline collocation methods for partial differential equations. J. Comput. Appl. Math. 128, 55–82 (2001). Numerical analysis 2000, vol. VII, Partial differential equations · Zbl 0971.65105 · doi:10.1016/S0377-0427(00)00509-4
[3] Chen, T.: An efficient algorithm based on quadratic spline collocation and finite difference methods for parabolic partial differential equations. Master’s thesis, University of Toronto, Toronto, Ontario, Canada (2005)
[4] Christara, C.C.: Quadratic spline collocation methods for elliptic partial differential equations. BIT 34, 33–61 (1994) · Zbl 0815.65118 · doi:10.1007/BF01935015
[5] Christara, C.C., Ng, K.S.: Adaptive techniques for spline collocation. Computing 76, 259–277 (2006) · Zbl 1086.65077 · doi:10.1007/s00607-005-0141-3
[6] Christara, C.C., Ng, K.S.: Optimal quadratic and cubic spline collocation on nonuniform partitions. Computing 76, 227–257 (2006) · Zbl 1086.65076 · doi:10.1007/s00607-005-0140-4
[7] Dang, D.M.: Adaptive finite difference methods for valuing American options. Master’s thesis, University of Toronto, Toronto, Ontario (2007)
[8] de Boor, C., Swartz, B.: Collocation at Gaussian points. SIAM J. Numer. Anal. 10, 582–606 (1973) · Zbl 0232.65065 · doi:10.1137/0710052
[9] Douglas, J., Dupont, T.: A finite element collocation method for quasilinear parabolic equations. Math. Comput. 27, 17–28 (1973) · Zbl 0256.65050 · doi:10.1090/S0025-5718-1973-0339508-8
[10] Douglas, J., Dupont, T.: Collocation methods for parabolic equations in a single-space variable. Lect. Notes Math. 385, 1–147 (1974) · Zbl 0279.65097 · doi:10.1007/BFb0057338
[11] Forsyth, P.A., Vetzal, K.: Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. Comput. 23, 2095–2122 (2002) · Zbl 1020.91017 · doi:10.1137/S1064827500382324
[12] Greenwell-Yanik, C.E., Fairweather, G.: Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables. SIAM J. Numer. Anal. 23, 282–296 (1986) · Zbl 0595.65122 · doi:10.1137/0723020
[13] Houstis, E.N., Christara, C.C., Rice, J.R.: Quadratic-spline collocation methods for two-point boundary value problems. Int. J. Numer. Methods Eng. 26, 935–952 (1988) · Zbl 0654.65057 · doi:10.1002/nme.1620260412
[14] Houstis, E.N., Rice, J.R., Christara, C.C., Vavalis, E.A.: Performance of scientific software. In: Rice, J.R. (ed.) The IMA Volumes in Mathematics and its Applications. Mathematical Aspects of Scientific Software, vol. 14, pp. 123–155 (1988) · Zbl 0683.65098
[15] Hull, J.C.: Options, Futures, and Other Derivatives, 6th edn. Prentice Hall, Englewood Cliffs (2006) · Zbl 1087.91025
[16] Iserles, A.: A first Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1997) · Zbl 0886.65073
[17] Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numer. Math. 43, 309–327 (1984) · Zbl 0524.65072 · doi:10.1007/BF01390130
[18] Wang, R., Keast, P., Muir, P.: BACOL: B-Spline adaptive COLlocation software for 1-D parabolic PDEs. ACM Trans. Math. Softw. 30, 454–470 (2004) · Zbl 1070.65564 · doi:10.1145/1039813.1039817
[19] Wang, R., Keast, P., Muir, P.: A high-order global spatially adaptive collocation method for 1-D parabolic PDEs. Appl. Numer. Math. 50, 239–260 (2004) · Zbl 1049.65110 · doi:10.1016/j.apnum.2003.12.023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.