Topic
Discretization
About: Discretization is a research topic. Over the lifetime, 53069 publications have been published within this topic receiving 1077475 citations.
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TL;DR: This paper reviews and further develops a class of strong stability-preserving high-order time discretizations for semidiscrete method of lines approximations of partial differential equations, and builds on the study of the SSP property of implicit Runge--Kutta and multistep methods.
Abstract: In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.
2,199 citations
TL;DR: A class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in Shu& Osher (1988), suitable for solving hyperbolic conservation laws with stable spatial discretizations is explored, verifying the claim that TVD runge-kutta methods are important for such applications.
Abstract: In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in Shu& Osher (1988), suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.
2,146 citations
TL;DR: In this article, a spectral element method was proposed for numerical solution of the Navier-Stokes equations, where the computational domain is broken into a series of elements, and the velocity in each element is represented as a highorder Lagrangian interpolant through Chebyshev collocation points.
2,133 citations
09 Jul 1995
TL;DR: Binning, an unsupervised discretization method, is compared to entropy-based and purity-based methods, which are supervised algorithms, and it is found that the performance of the Naive-Bayes algorithm significantly improved when features were discretized using an entropy- based method.
Abstract: Many supervised machine learning algorithms require a discrete feature space. In this paper, we review previous work on continuous feature discretization, identify defining characteristics of the methods, and conduct an empirical evaluation of several methods. We compare binning, an unsupervised discretization method, to entropy-based and purity-based methods, which are supervised algorithms. We found that the performance of the Naive-Bayes algorithm significantly improved when features were discretized using an entropy-based method. In fact, over the 16 tested datasets, the discretized version of Naive-Bayes slightly outperformed C4.5 on average. We also show that in some cases, the performance of the C4.5 induction algorithm significantly improved if features were discretized in advance; in our experiments, the performance never significantly degraded, an interesting phenomenon considering the fact that C4.5 is capable of locally discretizing features.
2,089 citations
TL;DR: A continuous surface charge (CSC) approach is introduced that leads to a smooth and robust formalism for the PCM models and achieves a clear separation between "model" and "cavity" which, together with simple generalizations of modern integral codes, is all that is required for an extensible and efficient implementation of thePCM models.
Abstract: Continuum solvation models are appealing because of the simplified yet accurate description they provide of the solvent effect on a solute, described either by quantum mechanical or classical methods. The polarizable continuum model (PCM) family of solvation models is among the most widely used, although their application has been hampered by discontinuities and singularities arising from the discretization of the integral equations at the solute-solvent interface. In this contribution we introduce a continuous surface charge (CSC) approach that leads to a smooth and robust formalism for the PCM models. We start from the scheme proposed over ten years ago by York and Karplus and we generalize it in various ways, including the extension to analytic second derivatives with respect to atomic positions. We propose an optimal discrete representation of the integral operators required for the determination of the apparent surface charge. We achieve a clear separation between “model” and “cavity” which, together with simple generalizations of modern integral codes, is all that is required for an extensible and efficient implementation of the PCM models. Following this approach we are now able to introduce solvent effects on energies, structures, and vibrational frequencies (analytical first and second derivatives with respect to atomic coordinates), magnetic properties (derivatives with respect of magnetic field using GIAOs), and in the calculation more complex properties like frequency-dependent Raman activities, vibrational circular dichroism, and Raman optical activity.
2,033 citations