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High-order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions
journal contribution
posted on 2023-06-08, 22:23 authored by Bertram Duering, Christof HeuerWe 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.
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Publication status
- Published
File Version
- Published version
Journal
SIAM Journal on Numerical AnalysisISSN
0036-1429Publisher
Society for Industrial and Applied MathematicsExternal DOI
Issue
5Volume
53Page range
2113-2134Department affiliated with
- Mathematics Publications
Research groups affiliated with
- Numerical Analysis and Scientific Computing Research Group Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2015-09-04First Open Access (FOA) Date
2015-09-04First Compliant Deposit (FCD) Date
2015-09-03Usage metrics
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