Explicit Runge-Kutta methods for parabolic partial differential equations

Verwer, J. G.

doi:10.1016/s0168-9274(96)00022-0

Cited by 154 publications

(125 citation statements)

References 22 publications

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“…In our approach, different information coming from the linear solver itself is used for the stability control, and this is shown to be more efficient in practice. Unlike many other explicit stabilized methods (see, e.g., [23]), such as RKC, MRAI successfully copes with the complex spectrum Jacobian.…”

Section: Resultsmentioning

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%

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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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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Botchev

Sleijpen

Vorst

1999

Applied Numerical Mathematics

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“…One way of increasing the stability regions of Runge-Kutta integrators is to construct explicit Runge-Kutta integrators that have larger numbers of stages than the minimum required to satisfy integration order conditions and to select integrator coefficients in a way that enlarges the stability region. A number of researchers have worked on developing rigorous methodologies for doing this; e.g., Kinnmark and Gray (1984a, b), Sommeijer et al (1998), van der Houwen (1977, and Verwer (1996).…”

Section: Stabilized Runge-kutta Methodsmentioning

confidence: 99%

Dual-Rate Integration Using Partitioned Runge-Kutta Methods for Mechanical Systems with Interacting Subsystems

Shome

Haug

Jay

2004

Mechanics Based Design of Structures and Machines

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A framework is presented allowing dual-rate numerical integration of the equations of mechanical system dynamics to be considered as a form of Partitioned Runge-Kutta (PRK) integration. Certain coefficients of a PRK integrator are set to zero, so that Runge-Kutta integrators that constitute the PRK integrator can be made to have different numbers of stages. As a result, one Runge-Kutta integrator requires fewer function evaluations than the other does, which is a form of dual-rate integration. Well-established order of accuracy theory for PRK integrators is used to develop a rigorous methodology for designing explicit PRK dual-rate integrators. Stabilized Runge-Kutta theory developed for single-rate RungeKutta integrators is combined with PRK integrator theory to design PRK dual-rate integrators with the largest possible stability regions. Dual-rate PRK integrators created using these approaches are used to simulate the dynamics of vehicle systems that contain subsystems with higher frequency response characteristics than do the basic vehicle subsystems.

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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%

“…We wish to emphasize that, while the composite scheme is employed in this work for the purposes of stability analysis, its greater generality may prove it to be the appropriate choice for the numerical integration of yIt has been claimed by Verwer (1996) that factorized RKC methods are impractical as they suffer from severe internal instability. We find no evidence of this for N STS 9 30.…”

Section: Compositementioning

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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Explicit Runge-Kutta methods for parabolic partial differential equations

Cited by 154 publications

References 22 publications

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

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

Dual-Rate Integration Using Partitioned Runge-Kutta Methods for Mechanical Systems with Interacting Subsystems

On the acceleration of explicit finite difference methods for option pricing

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