Journal of Nonlinear and Variational Analysis

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DOI: 10.23952/jnva.9.2025.1.03

Volume 9, Issue 1, 1 February 2025, Pages 31-43

 

Abstract. In this paper, we propose two linear-implicit local energy dissipation-preserving algorithms for a gradient flow system. We first prove that the gradient flow system possesses a local energy dissipation law, which is exactly conserved within any local time-space region. We then introduce an auxiliary variable to reformulate the gradient flow system into an equivalent system, which is proven to preserve the local energy dissipation property. To maintain the intrinsic properties as many as possible, two linear-implicit local energy dissipation-preserving algorithms are developed by means of the composition method. Furthermore, we prove that the proposed algorithms adhere to the discrete local energy dissipation laws with the assistance of the Leibnitz rules. Particularly, under appropriate boundary conditions, these innovative algorithms naturally preserve the discrete total mass laws and ensure the global energy stability in the sense of energy decay for the gradient flows. Finally, numerical examples are provided to demonstrate the efficiency of the proposed algorithms and their effectiveness in preserving the energy dissipation laws.

 

How to Cite this Article:
Z. Mu, Y. Wang, Linear-implicit local energy dissipation-preserving algorithms for the gradient flow system, J. Nonlinear Var. Anal. 9 (2025), 31-43.