The midpoint upwind scheme

Stynes, Martin; Roos, Hans–Görg

doi:10.1016/s0168-9274(96)00071-2

Cited by 128 publications

(50 citation statements)

References 6 publications

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“…Our hybrid difference scheme uses central differences whenever the local mesh size is small enough to ensure the stability of the scheme; otherwise we use a simple upwind difference scheme, which only occurs at the first grid point. The scheme is a modification of the difference scheme used in [14] and [15]. The scheme is stable for arbitrary volatility and arbitrary interest rate.…”

Section: Resultsmentioning

confidence: 99%

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A robust finite difference scheme for pricing American put options with Singularity-Separating method

Cen

2009

Numer Algor

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In this paper we present a stable numerical method for the linear complementary problem arising from American put option pricing. The numerical method is based on a hybrid finite difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique. The scheme is stable for arbitrary volatility and arbitrary interest rate. We apply some tricks to derive the error estimates for the direct application of finite difference method to the linear complementary problem. We use the Singularity-Separating method to remove the singularity of the non-smooth payoff function. It is proved that the scheme is second-order convergent with respect to the spatial variable. Numerical results support the theoretical results.

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Section: Resultsmentioning

confidence: 99%

“…We discretize the Black-Scholes operator using a hybrid finite difference scheme on the above piecewise-uniform mesh. Our discretization is similar to that of [14] and [15] in that it uses the central difference approximation…”

Section: The Hybrid Finite Difference Schemementioning

confidence: 99%

A robust finite difference scheme for pricing American put options with Singularity-Separating method

Cen

2009

Numer Algor

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“…Using grid equidistribution, Beckett and Mackenzie [9] constructed a method in order near 2. Stynes and Roos [10] proposed a difference scheme, which is uniformly convergent with order near 2. A streamline diffusion method, which is uniformly convergent with order 2, is presented also by Stynes and Tobiska [11] .…”

Section: Introductionmentioning

confidence: 99%

High accuracy non-equidistant method for singular perturbation reaction-diffusion problem

Cai

Rui-qian

et al. 2009

Appl. Math. Mech.-Engl. Ed.

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Singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region. A non-equidistant finite difference method is presented according to the property of boundary layer. The region is divided into an inner boundary layer region and an outer boundary layer region according to transition point of Shishkin. The steps sizes are equidistant in the outer boundary layer region. The step sizes are gradually increased in the inner boundary layer region such that half of the step sizes are different from each other. Truncation error is estimated. The proposed method is stable and uniformly convergent with the order higher than 2. Numerical results are given, which are in agreement with the theoretical result.

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“…These schemes can be easily extended into two dimensions (unlike, e.g., three-point second-order schemes like [2,18]). Note also that a similar many-point regularization idea leads, e.g., to the Gontcharov-Frjasinov five-point scheme [5], which works well for the Navier-Stokes equations at high Reynolds numbers.…”

Section: Introductionmentioning

confidence: 99%

Uniform Pointwise Convergence of Difference Schemes for Convection-Diffusion Problems on Layer-Adapted Meshes

Kopteva¹

2001

Computing

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We consider two convection-diffusion boundary value problems in conservative form: for an ordinary differential equation and for a parabolic equation. Both the problems are discretized using a four-point second-order upwind space difference operator on arbitrary and layer-adapted space meshes. We give ε-uniform maximum norm error estimates O N −2 ln 2 N ( + τ ) and O N −2 ( + τ ) , respectively, for the Shishkin and Bakhvalov space meshes, where N is the space meshnodes number, τ is the time meshinterval. The smoothness condition for the Bakhvalov mesh is replaced by a weaker condition.

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The midpoint upwind scheme

Cited by 128 publications

References 6 publications

A robust finite difference scheme for pricing American put options with Singularity-Separating method

A robust finite difference scheme for pricing American put options with Singularity-Separating method

High accuracy non-equidistant method for singular perturbation reaction-diffusion problem

Uniform Pointwise Convergence of Difference Schemes for Convection-Diffusion Problems on Layer-Adapted Meshes

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