RKC: An explicit solver for parabolic PDEs

Sommeijer, B. P.; Shampine, L. F.; Verwer, J. G.

doi:10.1016/s0377-0427(97)00219-7

Cited by 163 publications

(197 citation statements)

References 12 publications

(27 reference statements)

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“…In the following, we shall use the description of Alexiades et al (1996), itself a variant of a method presented by Gentzsch (1979) and essentially a pareddown Runge-Kutta-Chebyshev (RKC) method (van der Houwen 1977, van der Houwen and Sommeijer 1980, Verwer et al 1990, Verwer 1996, Sommeijer et al 1997.…”

Section: Super-time-steppingmentioning

confidence: 99%

On the acceleration of explicit finite difference methods for option pricing

O’Sullivan

2011

Quantitative Finance

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Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant-Friedrichs-Lewy (CFL) condition which limits the latter class's efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black-Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes.Numerical methods for option pricing, Black-Scholes model, Computational finance, Equity options, American options, Exotic options,

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Section: Super-time-steppingmentioning

confidence: 99%

On the acceleration of explicit finite difference methods for option pricing

O’Sullivan

2011

Quantitative Finance

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“…In fact, condition ( 17) guarantees that the rightmost local maximum of Q (µ) ( cf. The fact that matrix A is diagonal does not affect behavior of the minimum residual iterations.…”

Section: Over the Interval Of Interest We Have Thatmentioning

confidence: 99%

“…The conditions (17) and (26) have been derived for a negative real spectrum of A. It turned out experimentally that when k ;;::: 3 they may also be used when the spectrum of A is contained in the left complex half-plane.…”

Section: -µmentioning

confidence: 99%

“…This test problem has been taken from [17]; it is a linear heat conduction problem over the 30 unit cube. The inhomogeneous tem1 is chosen in order to have the analytical solution tanh(5(x + 2y + 1.5z -0.5 -t)).…”

Section: D Heat Conduction Problemmentioning

confidence: 99%

“…If one uses k iterative steps, then the resulting approximate method may be viewed as an explicit integration scheme and such schemes have been studied heavily. Most often, these schemes are studied with fixed coefficients of Chebyshev approximations [11,12,16,17,21,23]. Performance of these methods depends on a priori knowledge of a region containing the spectrum of the Jacobian.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability control for approximate implicit time stepping schemes with minimum residual iterations

Botchev

Sleijpen

Vorst

1999

Applied Numerical Mathematics

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Implicit schemes for the integration of ODEs are popular when stability is more of concern than accuracy, for instance for the computation of a steady state solution. However, in particular for very large systems, the solution of the involved linear systems may be very expensive. When these systems are solved iteratively to a certain tolerance, it is often not known which tolerance has to be taken. We propose a different strategy, where the number of iterations is fixed, but the step size is controlled with respect to stability. Numerical tests show the effectiveness of this approach in comparison with an implicit scheme that iterates to a certain tolerance.

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Large stiff systems solved by Chebyshev methods

Abdulle

2002

Proc. Appl. Math. Mech.

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Large stiff systems solved by Chebyshev methodsChebyshev methods (also called stabilized methods) are explicit Runge-Kutta methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. We present here new Chebyshev methods of second and fourth order called ROCK, which can be seen as a combination and a generalization of van der Houwen-Sommeijer-type methods and Lebedev-type methods.

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RKC: An explicit solver for parabolic PDEs

Cited by 163 publications

References 12 publications

On the acceleration of explicit finite difference methods for option pricing

On the acceleration of explicit finite difference methods for option pricing

Stability control for approximate implicit time stepping schemes with minimum residual iterations

Large stiff systems solved by Chebyshev methods

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