High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions

25 Pages Posted: 28 Jun 2014

See all articles by Bertram Düring

Bertram Düring

University of Warwick - Mathematics Institute

Christof Heuer

University of Sussex - School of Mathematical and Physical Sciences

Date Written: June 27, 2014

Abstract

We present a high-order compact finite difference approach for a rather general class of parabolic partial differential equations with time and space dependent coefficients as well as with mixed second-order derivative terms in n spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance. We derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover, we perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example we consider the pricing of European Power Put Options in the multidimensional Black-Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions we employ the smoothing operators of Kreiss et al. to ensure high-order convergence of the approximations of the smoothed problem to the true solution.

Keywords: High-order compact scheme, parabolic PDE, basket options, mixed derivatives, stability

JEL Classification: C63, G13

Suggested Citation

Düring, Bertram and Heuer, Christof, High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions (June 27, 2014). Available at SSRN: https://ssrn.com/abstract=2459861 or http://dx.doi.org/10.2139/ssrn.2459861

Bertram Düring (Contact Author)

University of Warwick - Mathematics Institute ( email )

Zeeman Building
Coventry, CV4 7AL
United Kingdom

Christof Heuer

University of Sussex - School of Mathematical and Physical Sciences ( email )

Brighton, BN1 9QH
United Kingdom

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