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Smoothing with positivity‐preserving Padé schemes for parabolic problems with nonsmooth data
Author(s) -
Wade B. A.,
Khaliq A. Q. M.,
Siddique M.,
Yousuf M.
Publication year - 2005
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20039
Subject(s) - mathematics , hilbert space , smoothing , partial differential equation , convergence (economics) , partial derivative , class (philosophy) , operator (biology) , diagonal , mathematical optimization , mathematical analysis , computer science , biochemistry , statistics , chemistry , geometry , repressor , artificial intelligence , transcription factor , economics , gene , economic growth
We introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well‐known Rannacher schemes. The new family of algorithms utilizes diagonal Padé schemes combined with positivity‐preserving Padé schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self‐adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005