Development of circular arc boundary elements method

Dehghan, Mehdi; Hosseinzadeh, Hossein

doi:10.1016/j.enganabound.2010.08.014

Cited by 11 publications

(6 citation statements)

References 8 publications

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“…(8) In LRPI method, using the delta Kronecker property, the boundary conditions can be easily imposed. (9) The optimal values of the size of local sub-domain and support domain (r Q and r w ) are illustrated using the Figures 4 and 5 (11) Future work will concern an extension to the basket options.…”

Section: Discussionmentioning

confidence: 99%

“…To overcome this difficulty associated with FEM and FVM, the boundary element method (BEM) [6] appears to be a attractive alternative. In the BEM, only the boundary of domain needs to be discretized [7,8,9,10]. This reduces the problem dimension by one and thus largely reduces the time in meshing.…”

Section: Introductionmentioning

confidence: 99%

“…This reduces the problem dimension by one and thus largely reduces the time in meshing. The BEM still uses elements to implement both interpolation and integration [7,8,9,10].…”

Section: Introductionmentioning

confidence: 99%

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Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov-Galerkin method

Rad¹,

Parand²

2014

Preprint

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The most recent update of financial option models is American options under stochastic volatility models with jumps in returns (SVJ) and stochastic volatility models with jumps in returns and volatility (SVCJ). To evaluate these options, mesh-based methods are applied in a number of papers but it is well-known that these methods depend strongly on the mesh properties which is the major disadvantage of them. Therefore, we propose the use of the meshless methods to solve the aforementioned options models, especially in this work we select and analyze one scheme of them, named local radial point interpolation (LRPI) based on Wendland's compactly supported radial basis functions (WCS-RBFs) with C 6 , C 4 and C 2 smoothness degrees. The LRPI method which is a special type of meshless local Petrov-Galerkin method (MLPG), offers several advantages over the mesh-based methods, nevertheless it has never been applied to option pricing, at least to the very best of our knowledge. These schemes are the truly meshless methods, because, a traditional non-overlapping continuous mesh is not required, neither for the construction of the shape functions, nor for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the implicit-explicit (IMEX) time stepping scheme is employed for the time derivative which allows us to smooth the discontinuities of the options' payoffs. Stability analysis of the method is analyzed and performed. In fact, according to an analysis carried out in the present paper, the proposed method is unconditionally stable. Numerical experiments are presented showing that the proposed approaches are extremely accurate and fast.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov-Galerkin method

Rad¹,

Parand²

2014

Preprint

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“…Meshless methods have also been developed for boundary integral equations to extend boundary type methods in which only the boundary of the domain of the problem need to be presented with nodes such as boundary node method (BNM) [53], local boundary integral equation method (LBIEM) [64], boundary cloud method (BCM) [47], boundary point method (BPM) [48], and the method of fundamental solution (MFS). The meshless methods require neither domain discretization as in finite element method (FEM) and finite difference method (FDM) [16], nor boundary discretization as in boundary element method (BEM) [17]. Thus they improve the computational efficiency and can be easily extended to solve high-ordered and high-dimensional differential equations.…”

Section: Introductionmentioning

confidence: 99%

A boundary-only meshless method for numerical solution of the Eikonal equation

Dehghan

Salehi

2010

Comput Mech

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The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method.

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“…The same method is also applied to find the solution of an hyperbolic equation [6]. In [7] the circular arc element implementation is compared to both linear and quadratic BEM discretizations. The DQM, which approximates the solution of the problem on a finite dimensional space by using polynomials as the basis of the space is applied to the nonlinear reaction-diffusion equation in one and two-space dimensions in Meral and Tezer-Sezgin's study [8], and the comparison of three different time integration methods (FDM, FEM, LSM) is made for the DQM solution of the one-dimensional nonlinear reaction-diffusion and wave equations in [9].…”

Section: Introductionmentioning

confidence: 99%