Scott Sarra

• PhD
• Professor at Marshall University

Radial basis function (RBF) methods are popular methods for scattered data interpolation and for solving PDEs in complexly shaped domains. RBF methods are simple to implement as they only require elementary linear algebra operations. In this work, center locations that result in matrices with a centrosymmetric structure are examined. The resulting...

Radial Basis Function (RBF) methods have become important tools for scattered data interpolation and for solving partial differential equations (PDEs) in complexly shaped domains. When the underlying function is sufficiently smooth, RBF methods can produce exceptional accuracy. However, like other high order numerical methods, if the underlying fun...

Radial Basis Function (RBF) methods are important tools for scattered data interpolation and for the solution of Partial Differential Equations in complexly shaped domains. The most straight forward approach used to evaluate the methods involves solving a linear system which is typically poorly conditioned. The Matlab Radial Basis Function toolbox...

Radial Basis Function (RBF) methods are important tools for scattered data interpolation and for the solution of PDEs in complexly shaped domains. Several approaches for the evaluation of RBF methods are known. To date, the most noteworthy methods are solving a linear system in the standard RBF basis using both double and extended precision floatin...

Scattered data interpolation using Radial Basis Functions involves solving an ill-conditioned symmetric positive definite (SPD) linear system (with appropriate selection of basis function) when the direct method is used to evaluate the problem. The standard algorithm for solving a SPD system is a Cholesky factorization. Severely ill-conditioned the...

Time-dependent advection–diffusion–reaction and diffusion–reaction equations are used as models in biology, chemistry, physics, and engineering. As representative examples, we focus on a chemotaxis model and a Turing system from biology and apply a local radial basis function method to numerically approximate the solutions. The numerical method can...

Gaussian radial basis function (RBF) interpolation methods are theoretically spectrally accurate. However, in applications this accuracy is seldom realized due to the necessity of solving a very poorly conditioned linear system to evaluate the methods. Recently, by using approximate cardinal functions and restricting the method to a uniformly space...

We examine the stability properties of a predictor-corrector implementation of a class of implicit linear multistep methods. The method has recently been described in the literature as suitable for the efficient integration of stiff systems and as having stability regions similar to well known implicit methods. A more detailed analysis reveals that...

Multiple results in the literature exist that indicate that all computed solutions to chaotic dynamical systems are time-step dependent. That is, solutions with small but different time steps will decouple from each other after a certain (small) finite amount of simulation time. When using double precision floating point arithmetic time step indepe...

Radial Basis Function (RBF) collocation methods for time-dependent PDEs, in particular hyperbolic PDEs, are known to be difficult to implement in a way so that they are stable for time integration. It has been hypothesized that the instability is due to the way that boundary conditions are applied and to the relatively large errors in boundary regi...

Radial basis function (RBF) methods that employ infinitely differentiable basis functions featuring a shape parameter are theoretically spectrally accurate methods for scattered data interpolation and for solving partial differential equations. It is also theoretically known that RBF methods are most accurate when the linear systems associated with...

Global polynomial approximation methods applied to piecewise continuous functions exhibit the well-known Gibbs phenomenon. We summarize known methods to remove the Gibbs oscillations and present a collection of Matlab programs that implement the methods. The software features a Graphical User Interface that allows easy access to the postprocessing...

Several variable shape parameter methods have been successfully used in Radial Basis Function approximation methods. In many cases variable shape parame-ter strategies produced more accurate results than if a constant shape parameter had been used. We introduce a new random variable shape parameter strat-egy and give numerical results showing that...

Pseudospectral Methods based on global polynomial approximation yield exponential accuracy when the underlying function is analytic. The presence of discontinuities destroys the extreme accuracy of the methods and the well-known Gibbs phenomenon appears. Several types of postprocessing methods have been developed to lessen the effects of the Gibbs...

This work focuses on the generalized multiquadric (GMQ) radial basis function. The GMQ is derived from the multiquadric (MQ), which is used in radial basis function (RBF) interpolation. This is a relatively new field of research, and many properties of the GMQ are still unknown. Numerical experiments will be performed involving the GMQ, and results...

Research monograph on the Multiquadric Radial Basis Function for the numerical solution of partial differential equations.

Under the governing equations of Hyperbolic Heat Transfer thermal disturbances travel with a finite speed of propagation and are visible as sharp discontinuities in the solution profiles. Due to the well-known Gibbs phenomenon, the numerical solution of hyperbolic heat trans-fer problems by high order numerical methods such as pseudospectral method...

Under the governing equations of hyperbolic heat transfer, thermal disturbances travel with a finite speed of propagation and are visible as sharp discontinuities in the solution profiles. As a result of the well-known Gibbs phenomenon, the numerical solution of hyperbolic heat transfer problems by high-order numerical methods such as pseudospectra...

Differentiation matrices associated with radial basis function (RBF) collocation methods often have eigenvalues with positive real parts of significant magnitude. This prevents the use of the methods for time-dependent problems, particulary if explicit time integration schemes are employed. In this work, accuracy and eigenvalue stability of symmetr...

Digital total variation (DTV) filtering techniques, that originated in the field of image processing, are adapted to postprocess radial basis function approximations of piecewise continuous functions. Through numerical examples, we show that DTV filtering is a fast, robust, postprocessing method that can be used to remove Gibbs oscillations while s...

Promising numerical results using once and twice integrated radial basis functions have been recently presented. In this work we investigate the integrated radial basis function (IRBF) concept in greater detail, connect to the existing RBF theory, and make conjectures about the properties of IRBF approximation methods. The IRBF methods are used to...

Digital total variation filtering is analyzed as a fast, robust, post-processing method for accelerating the convergence of pseudospectral approximations that have been contaminated by Gibbs oscillations. The method, which originated in image processing, can be combined with spectral filters to quickly post-process large data sets with sharp resolu...

Radial basis function (RBF) methods have shown the potential to be a universal grid free method for the numerical solution of partial differential equations. Both global and compactly supported basis functions may be used in the methods to achieve a higher order of accuracy. In this paper, we take advantage of the grid free property of the methods...

A software suite written in the Java programming language for the postprocessing of Chebyshev approximations to discontinuous functions is presented. It is demonstrated how to use the package to remove the effects of the Gibbs-Wilbraham phenomenon from Chebyshev approximations of discontinuous functions. Additionally, the package is used to postpro...

The numerical solution of a model describing a two-dimensional fluidized bed by a Chebyshev super spectral viscosity (SSV) method is considered. The model is in the form of a hyperbolic system of conservation laws with a source term, coupled with an elliptic equation for determining a stream function. The coupled elliptic equation is solved by a fi...

Under the governing equations of hyperbolic heat transfer, energy propagates through a medium as a wave with sharp discontinuities at the wave front. The use of spectral methods to solve such problems numerically results in a solution in which strong numerical oscillations are present due to the Gibbs-Wilbraham phenomenon. It is demonstrated that a...

A Chebyshev super spectral viscosity method and operator splitting are used to solve a hyperbolic system of conservation laws with a source term modeling a fluidized bed. The fluidized bed displays a slugging behavior which corresponds to shocks in the solution. A modified Gegenbauer postprocessing procedure is used to obtain a solution which is fr...

In this article and its accompanying applet, I introduce the method of characteristics for solving first order partial differential equations (PDEs). First, the method of characteristics is used to solve first order linear PDEs. Next, I apply the method to a first order nonlinear problem, an example of a conservation law, and I discuss why the meth...

The Rabinovich-Fabrikant system is a chaotic system of nonlinear ordi-nary differential equations in three dimensions. Using the Local Iterative Linearization method and a Runge-Kutta method (both of fourth order and identical step-size) phase plots are generated and compared. Issues concerning the numerical approximation of chaotic systems are exp...