Finite Volume Methods for Two-Dimensional Nonlocal Partial Differential Equation with a Gradient Flow Structure
-
Wenli Cai
,
Boqin Wei
Abstract
This paper delves into a class of nonlinear nonlocal partial differential equations, characterized by a gradient flow structure, as previously outlined in the literature. Finite volume methods are employed to investigate the numerical solutions of this model in two-dimensional settings. It reveals that the semi-discrete numerical scheme satisfies the entropy dissipation and the fully discrete numerical scheme preserves the positivity through a meticulous combination of numerical scheme definitions, upwind numerical fluxes and sophisticated estimation techniques. Furthermore, a novel contribution is made by demonstrating that the semi-discrete numerical scheme preserves the positivity and the fully discrete numerical scheme satisfies the entropy dissipation. These fundamental properties, pivotal in guaranteeing the convergence of numerical solutions towards steady states, are rigorously proven utilizing the invariant region method, complemented by detailed norm estimations and rigorous analytical techniques. Two-dimensional numerical experiments validate the entropy dissipation and the long-time convergence of numerical solutions generated by the scheme. The numerical analysis methodology and computational findings in this work offer valuable insights for research on models with gradient flow structures.
Funding source: Fundamental Research Funds for the Central Universities
Award Identifier / Grant number: 2021YQLX01
Funding statement: Cai was supported partially by the Fundamental Research Funds for the Central Universities (No. 2021YQLX01), 2025 Basic Sciences Initiative in Mathematics and Physics, Open Funds of State Key Laboratory for Geomechanics and Deep Underground Engineering (No. XD2025007), and Deep Earth Probe and Mineral Resources Exploration-National Science and Technology Major Project (No. 2024ZD1003902). The numerical calculations in this paper have been done on the supercomputing system in the Mathematics Laboratory of China University of Mining and Technology-Beijing.
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