J_ID: z8x Customer A_ID: 1170 Cadmus Art: NUM20228 — 2007/2/3 — page 1 — #1 High Order Smoothing Schemes for Inhomogeneous Parabolic Problems with Applications in Option Pricing A. Q. M. Khaliq, 1 B. A. Wade, 2 M. Yousuf, 3 J. Vigo-Aguiar 4 1 Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, Tennessee 37132-0001 2 Department of Mathematical Sciences, University of Wisconsin–Milwaukee, Milwaukee, Wisconsin 53201-0413 3 Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dahran, Saudi Arabia 4 Departamento de Matemática Aplicada, Universidad de Salamanca, 37003 Salamanca, Spain Received 25 May 2006; accepted 7 December 2006 Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.20228 A new family of numerical schemes for inhomogeneous parabolic partial differential equations is developed utilizing diagonal Padé schemes combined with positivity–preserving Padé schemes as damping devices. We also develop a split version of the algorithm using partial fraction decomposition to address difficul- ties with accuracy and computational efficiency in solving and to implement the algorithms in parallel. Numerical experiments are presented for several inhomogeneous parabolic problems, including pricing of financial options with nonsmooth payoffs. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 00: 000–000, 2007 Keywords: Black-Scholes PDE; nonsmooth data; nonsmooth pay-off; Padé Scheme; parabolic problem; positivity I. INTRODUCTION AQ1 For homogeneous parabolic partial differential equations (PDE) with nonsmooth initial data, a family of high order accurate smoothing schemes has recently been developed by Wade et al. [1]. Correspondence to: B. A. Wade, Department of Mathematical Sciences, University of Winconsin-Milwaukee, Milwaukee, WI 53201-0413 (email: wade@uwm.edu) Contract grant sponsor: Ministerio de Educatión, Cultura y Deporte, Spain; contract grant number: SAB2003-0266 Contract grant sponsor: U.S. National Security Agency; contract grant number: H98230-05-01-0062 Contract grant sponsor: Ministerio de Ciencia Technología; contract grant number: MTM2004-00295 Contract grant sponsor: Junta Castilla Lesn; contract grant number: SA024104 Contract grant sponsor: Middle Tennessee State University, Murfreesboro, Tennessee (FRC grant) © 2007 Wiley Periodicals, Inc.