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Meshfree methods
About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.
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TL;DR: In this article, Wu et al. proposed the local weak form meshless methods for option pricing; especially in this paper, they select and analyze two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI), and they use a powerful iterative algorithm named the Bi-conjugate gradient stabilized method (BCGSTAB) to get rid of this system.
54 citations
TL;DR: In this paper, a one-dimensional mesh-free particle formulation is proposed for simulating shock waves, which are associated with discontinuous phenomena, based on Taylor series expansion in the piecewise continuous regions on both sides of a discontinuity.
Abstract: In this paper, a one-dimensional meshfree particle formulation is proposed for simulating shock waves, which are associated with discontinuous phenomena. This new formulation is based on Taylor series expansion in the piecewise continuous regions on both sides of a discontinuity. The new formulation inherits the meshfree Lagrangian and particle nature of SPH, and is a natural extension and improvement on the traditional SPH method and the recently proposed corrective smoothed particle method (CSPM). The formulation is consistent even in the discontinuous regions. The resultant kernel and particle approximations consist of a primary part similar to that in CSPM, and a corrective part derived from the discontinuity. A numerical study is carried out to examine the performance of the formulation. The results show that the new formulation not only remedies the boundary deficiency problem but also simulates the discontinuity well. The formulation is applied to simulate the shock tube problem and a 1-D TNT slab detonation. It is found that the proposed formulation captures the shock wave at comparatively lower particle resolution. These preliminary numerical tests suggest that the new meshfree particle formulation is attractive in simulating hydrodynamic problems with discontinuities such as shocks waves.
54 citations
TL;DR: In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered, and the radial basis functions are used for finding an approximation of the solution of the present problem.
Abstract: Nonlocal mathematical models appear in various problems of physics and engineering. In these models the integral term may appear in the boundary conditions. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. These kinds of problems have certainly been one of the fastest growing areas in various application fields. The presence of an integral term in a boundary condition can greatly complicate the application of standard numerical techniques. As a well-known class of meshless methods, the radial basis functions are used for finding an approximation of the solution of the present problem. Numerical examples are given at the end of the paper to compare the efficiency of the radial basis functions with famous finite-difference methods. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008
54 citations
TL;DR: The renormalized meshfree derivatives are studied, which improve the consistency of the original weighted particle methods, and a hybrid particle scheme is built using the Godunov method and is numerically compared to the weak renormalization scheme.
Abstract: This paper is devoted to the study of a new kind of meshfree scheme based on a new class of meshfree derivatives: the renormalized meshfree derivatives, which improve the consistency of the original weighted particle methods. The weak renormalized meshfree scheme, built from the weak formulation of general conservation laws, turns out to be $L^2$ stable under some geometrical conditions on the distribution of particles and some regularity conditions of the transport field. A time discretization is then performed by analogy with finite volume methods, and the $L^1$, $L^\infty$, and $BV$ stabilities of the obtained time discretized scheme are studied. From the same analogy with finite volume methods, a hybrid particle scheme is built using the Godunov method and is numerically compared to the weak renormalized scheme.
54 citations
TL;DR: In this paper, the sources of error in the traditional Delaunay triangle-based numerical integration are investigated, and two alternative numerical integration schemes are analyzed: a local approach in which nodal shape function supports are exactly decomposed into triangles and circle segments is shown not to give accurate enough results, while a stabilized nodal quadrature scheme is shown to render high levels of accuracy.
Abstract: In this paper, issues regarding numerical integration of the discrete system of equations arising from natural neighbour (natural element) Galerkin methods are addressed. The sources of error in the traditional Delaunay triangle-based numerical integration are investigated. Two alternative numerical integration schemes are analysed. First, a ‘local’ approach in which nodal shape function supports are exactly decomposed into triangles and circle segments is shown not to give accurate enough results. Second, a stabilized nodal quadrature scheme is shown to render high levels of accuracy, while resulting specially appropriate in a Natural Neighbour Galerkin approximation method. The paper is completed with several examples showing the performance of the proposed techniques. Copyright © 2004 John Wiley & Sons, Ltd.
53 citations