Iterative methods and high-order difference schemes for 2D elliptic problems with mixed derivative

Fournié, Michel; Karaa, Samir

doi:10.1007/bf02832060

Cited by 14 publications

(10 citation statements)

References 20 publications

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“…The case with µ = 0.5 is given special attention, and numerical experiments supporting our theoretical analysis are presented. The results of this article generalize those obtained in [17] and [18] for solving 2D elliptic and parabolic problems with constant coefficients, respectively.…”

Section: Introductionsupporting

confidence: 65%

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

Karaa

2006

Numerical Methods Partial

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We propose a 9-point fourth-order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two-level high-order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth-order accurate in space and second-or lower-order accurate in time depending on the choice of a weighted average parameter µ. Unconditional stability is proved for 0.5 ≤ µ ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high-order accuracy of the schemes are presented.

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Section: Introductionsupporting

confidence: 65%

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

Karaa

2006

Numerical Methods Partial

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“…The fourth-order compact finite difference formula for the central point ( ) involves the nearest eight neighboring mesh points with mesh spacing h which is given by Fourni'e (2006) as in…”

Section: Fig 1 Labeling Of the Nine Grid Pointsmentioning

confidence: 99%

“…Ghadimi et al (2013) presented a new forth order finite difference scheme to solve the steady Navier-Stokes equation in the form of vorticity-streamfunction. They extended Fourni'e's (2006) formulation and applied it to solve Navier-Stokes equation in the benchmark problem of flow in the driven cavity up to . This advanced method has been used in the current study to simulate steady and unsteady flow around square cylinder to examine the accuracy and performance of this scheme.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations of Flow around Square Cylinder Using an Iterative High Order Difference Scheme

Fard¹,

Ghadimi²,

Zamanian³

2013

JAC

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A time dependent two-dimensional numerical simulation of flow around a rectangular cylinder is conducted using a particular high order compact finite difference method. The current forth-order compact scheme is implemented for the simulation of flow around bluff bodies for the first time. This formulation offers a semiexplicit method for the solution of Navier-Stokes equations in stream function-vorticity form and based on a nine point difference scheme with uniform grid spacing. The main advantage of the suggested method is its fast convergence and high accuracy.The proposed numerical method combines the enhanced Fourni'e's forth order scheme with a semi-explicit formulation. By this combination, very accurate results can be obtained with a relatively coarse mesh in a short time. Uniform grids are applied with this high performance algorithm, and the accuracy of the scheme is proved using the benchmark problem of incompressible laminar flow around a rectangular cylinder at medium and high Reynolds numbers. Current results have good agreement with other numerical and experimental values. Generally, it is shown that, using the high order finite difference method demonstrates fairly competitive results for the current problem.

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“…Another layer of complexity is added when the anisotropic case is considered and mixed second-order derivative terms are present in the operator. Few works on high-order compact schemes address this problem, and either study constant coefficient problems [7] or specific equations [2].…”

Section: Introductionmentioning

confidence: 99%

High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions

Düring¹,

Heuer²

2015

SIAM J. Numer. Anal.

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We present a high-order compact finite difference approach for a class of parabolic partial differential equations with time and space dependent coefficients as well as with mixed secondorder 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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Iterative methods and high-order difference schemes for 2D elliptic problems with mixed derivative

Cited by 14 publications

References 20 publications

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

Numerical Simulations of Flow around Square Cylinder Using an Iterative High Order Difference Scheme

High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions

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