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Solano-López, PabloCorresponding AuthorSaavedra, JorgeAuthorMolina, RaulAuthor

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Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations

Publicated to:Algorithms. 17 (12): 564- - 2024-12-01 17(12), DOI: 10.3390/a17120564

Authors: Solano-Lopez, Pablo; Saavedra, Jorge; Molina, Raul

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Abstract

This work explores the numerical translation of the weak or integral solution of nonlinear partial differential equations into a numerically efficient, time-evolving scheme. Specifically, we focus on partial differential equations separable into a quasilinear term and a nonlinear one, with the former defining the Green function of the problem. Utilizing the Green function under a short-time approximation, it becomes possible to derive the integral solution of the problem by breaking it into three integral terms: the propagation of initial conditions and the contributions of the nonlinear and boundary terms. Accordingly, we follow this division to describe and separately analyze the resulting algorithm. To ensure low interpolation error and accurate numerical Green functions, we adapt a piecewise interpolation collocation method to the integral scheme, optimizing the positioning of grid points near the boundary region. At the same time, we employ a second-order quadrature method in time to efficiently implement the nonlinear terms. Validation of both adapted methodologies is conducted by applying them to problems with known analytical solution, as well as to more challenging, norm-preserving problems such as the Burgers equation and the soliton solution of the nonlinear Schr & ouml;dinger equation. Finally, the boundary term is derived and validated using a series of test cases that cover the range of possible scenarios for boundary problems within the introduced methodology.

Keywords

Advective equationBoundary conditionBoundary conditionsDistributed gridFokker-planck equationsGreen functionHeat transfer equationHigh-orderHydrodynamicsIntegral kernelIntegral methodNonlinearNumerical evaluationNumerical methodPath-integral solutionsSchemTime

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

The work has been published in the journal Algorithms due to its progression and the good impact it has achieved in recent years, according to the agency WoS (JCR), it has become a reference in its field. In the year of publication of the work, 2024 there are still no calculated indicators, but in 2023, it was in position 136/197, thus managing to position itself as a Q2 (Segundo Cuartil), in the category Computer Science, Artificial Intelligence. Notably, the journal is positioned en el Cuartil Q2 para la agencia Scopus (SJR) en la categoría Numerical Analysis.

Impact and social visibility

From the perspective of influence or social adoption, and based on metrics associated with mentions and interactions provided by agencies specializing in calculating the so-called "Alternative or Social Metrics," we can highlight as of 2025-06-08:

With a more dissemination-oriented intent and targeting more general audiences, we can observe other more global scores such as:

    It is essential to present evidence supporting full alignment with institutional principles and guidelines on Open Science and the Conservation and Dissemination of Intellectual Heritage. A clear example of this is:

    • The work has been submitted to a journal whose editorial policy allows open Open Access publication.

    Leadership analysis of institutional authors

    There is a significant leadership presence as some of the institution’s authors appear as the first or last signer, detailed as follows: First Author (Solano López, Pablo) and Last Author (Molina Gil, Raúl).

    the author responsible for correspondence tasks has been Solano López, Pablo.