The construction of hopscotch methods for parabolic and elliptic equations in two space dimensions with a mixed derivative

Gourlay, A. R.; McKee, Sean

doi:10.1016/s0377-0427(77)80009-5

Cited by 62 publications

(30 citation statements)

References 6 publications

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“…Though this is very inefficient to implement except in one dimension, so-called alternating direction implicit (ADI) schemes are much more practical and retain much of the theoretical stability. Variants of ADI which treat cross diffusion have been formulated by Beam and Warming (1980) and Gourlay and McKee (1977), but they have also been found vulnerable to unphysical, negative values. The explicit scheme could be ''fixed'' with a Lax approach, which replaces f n ij in the time difference ðf nþ1 ij À f n ij Þ=Dt with its average over neighboring grid points.…”

Section: Other Algorithmsmentioning

confidence: 99%

The coupling of quasi-linear pitch angle and energy diffusion

Albert

2009

Journal of Atmospheric and Solar-Terrestrial Physics

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Section: Other Algorithmsmentioning

confidence: 99%

The coupling of quasi-linear pitch angle and energy diffusion

Albert

2009

Journal of Atmospheric and Solar-Terrestrial Physics

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“…Test results of the line hopscotch method, written in its fast form and applied to non-autonomous equations, can be found in [13,21].…”

Section: The Line Hopscotch Methodsmentioning

confidence: 99%

One-step splitting methods for semi-discrete parabolic equations

Houwen¹,

Verwer²

1979

Computing

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--ZusammenfassungOne-Step Splitting Methods for Semi-Discrete Parabolic Equations. The main purpose of the paper is to discuss splitting methods for parabolic equations via the method of Iines. Firstly, we deal with the formulation of these methods for autonomous semi-discrete equations formeln wird hier definiert welche alle bekannten Split-Methoden entNilt, unter der Bedingung, dab die Funktionen f~ geschickt gew~ihlt sind. Fiir eine Reihe yon Methoden werden ~.tabilit~itsergebnisse angegeben. Ferner wird das alternierende Richtungsverfahren fiir Probleme mit einer wiltkiirlichen nicht-linearen Verbindung zwischen Raumableitungen diskutiert.

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“…It is seen that the direct finite difference discretization of the cross-derivative term in the two-asset option price equation would lead to an explicit scheme with 9 points at the old time level. On the other hand, the frequently used Hopscotch method [5] still involves 7 points at the old time level. The explicit finite difference scheme depicted below, which is derived using the Fourier method, uses a symmetric stencil which involves only 5 points at the old time level.…”

Section: Valuation Algorithms For Two-asset Option Modelsmentioning

confidence: 99%

Pricing Algorithms of Multivariate Path Dependent Options

Kwok

Wong

Lau

2001

Journal of Complexity

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Financial derivatives which are multivariate in nature are abundant in the financial markets. The underlying state variables may be the stock prices, interest rates, exchange rates, stochastic volatility, average of stock prices, extremum values of stock priors, etc. Option contracts whose life and payoff depend on the stochastic movement of the underlying asset prices are termed path dependent options. In this paper, we examine the pricing methods of several prototype path dependent options. These include options with sequential barriers, options with an external barrier and two-asset lookback options. The governing equations for the option prices are seen to resemble the diffusion type equations but with cross derivative terms, a feature which differs from the usual diffusion equations in engineering. Various techniques to reduce the complexity of the multivariate nature of these prototype option pricing models are discussed. It is illustrated that the dimensionality of a path dependent option model may be reduced by some ingenious choices of similarity variables. We also examine the design of pricing algorithms of these multivariate options, in particular, with regard to the treatment of discrete monitoring feature and the prescription of numerical boundary conditions. The possible generalizations of the numerical techniques presented in this paper to other models with more complicated path dependent payoff structures are also discussed.

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The construction of hopscotch methods for parabolic and elliptic equations in two space dimensions with a mixed derivative

Cited by 62 publications

References 6 publications

The coupling of quasi-linear pitch angle and energy diffusion

The coupling of quasi-linear pitch angle and energy diffusion

One-step splitting methods for semi-discrete parabolic equations

Pricing Algorithms of Multivariate Path Dependent Options

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