Construction of the radial basis function finite difference methods and their application to problems with arbitrary domain
A wide variety of problems in mathematics and particularly in computational mathematics is formulated as second-order partial differential equations (PDEs), which mostly do not admit closed-form solutions. Due to this, constructing numerical solver for such problems is requisite as well as of special importance when the computational domain is not anymore regular. Such problems, which mainly occur in engineering modeling problems, are defined on irregular domain. Classic methods such as finite difference (FD) or spectral solvers are hard to be employed on irregular domains with arbitrary geometries. One remedy is to rely on finite element method or the meshless radial basis function (RBF) method. This is discussed in this paper. Toward this purpose, an RBF-FD method is discussed and its weighting coefficients are constructed theoretically. Numerical reports and comparison will confirm the efficacy and competitiveness of the presented approach on domains with arbitrary geometries.
The meshfree RBF methods are categorized as globalized and localized meshless schemes. A localized version of such schemes is the RBF-FD estimates, which is an attractive choice in contrast to the global meshfree RBF schemes by providing better-conditioned and sparse discretization matrices. This clearly reduces the computational load of applying RBF methods for practical problems in higher dimensions. Unlike the standard procedures, the RBF-FD method can deal with scattered node layouts and irregular domains. In addition, their locality makes them more flexible with respect to local refinement techniques than the globalized RBF methods.
In this paper, we obtain weights of the radial basis function finite difference formula for some differential operators. These weights are used to obtain the local truncation error in function derivative approximating. We discuss how to select the shape parameter for the RBF-FD formulas so as to gain the accuracy as well as stability. We apply these formulas for Poisson equation with irregular domains and show that the proposed method can be used as a fully meshfree method.
The following conclusions were drawn from this research. Closed-form coefficients for RBF-FD formulas to approximate derivatives of the function can be obtained under some conditions. It is discussed that how we can choose the shape parameters to have efficient RBF-FD formulations in solving PDE problems on irregular domains. Several experiments on different types of irregular geometries are discussed and compared and showed how much RBF-FD is useful for practical engineering PDE problems on irregular domains.
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Finding an efficient machine learning predictor for lesser liquid credit default swaps in equity markets
F. Soleymani *
Iranian Journal of Numerical Analysis and Optimization, Winter 2023