Authors:Xiaoli Li, Jie Shen Pages: 2263 - 2297 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 12, Page 2263-2297, November 2020. We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-10-19T07:00:00Z DOI: 10.1142/S0218202520500438 Issue No:Vol. 30, No. 12 (2020)
Authors:Fei Jiang, Song Jiang, Weicheng Zhan Pages: 2299 - 2388 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 12, Page 2299-2388, November 2020. Based on a bootstrap instability method, we prove the existence of unstable strong solutions in the sense of [math]-norm to an abstract Rayleigh–Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates. In the proof we develop a method to modify the initial data of the linearized abstract RT problem by exploiting the existence theory of a unique solution to the stratified (steady) Stokes problem and an iterative technique, such that the obtained modified initial data satisfy the necessary compatibility conditions on boundary of the original (nonlinear) abstract RT problem. Applying an inverse transform of Lagrangian coordinates to the obtained unstable solutions and taking then proper values of the parameters, we can further obtain unstable solutions of the RT problem in viscoelastic, magnetohydrodynamics (MHD) flows with zero resistivity and pure viscous flows (with/without interface intension) in Eulerian coordinates. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-10-23T07:00:00Z DOI: 10.1142/S021820252050044X Issue No:Vol. 30, No. 12 (2020)
Authors:Piotr Minakowski, Piotr B. Mucha, Jan Peszek Pages: 2389 - 2415 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 12, Page 2389-2415, November 2020. The paper introduces a model of collective behavior where agents receive information only from sufficiently dense crowds in their immediate vicinity. The system is an asymmetric, density-induced version of the Cucker–Smale model with short-range interactions. We prove the basic mathematical properties of the system and concentrate on the presentation of interesting behaviors of the solutions. The results are illustrated by numerical simulations. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-10-23T07:00:00Z DOI: 10.1142/S0218202520500451 Issue No:Vol. 30, No. 12 (2020)
Authors:Seung-Yeal Ha, Shi Jin, Doheon Kim Pages: 2417 - 2444 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 12, Page 2417-2444, November 2020. Global optimization of a non-convex objective function often appears in large-scale machine learning and artificial intelligence applications. Recently, consensus-based optimization (CBO) methods have been introduced as one of the gradient-free optimization methods. In this paper, we provide a convergence analysis for the first-order CBO method in [J. A. Carrillo, S. Jin, L. Li and Y. Zhu, A consensus-based global optimization method for high dimensional machine learning problems, https://arxiv.org/abs/1909.09249v1]. Prior to this work, the convergence study was carried out for CBO methods on corresponding mean-field limit, a Fokker–Planck equation, which does not imply the convergence of the CBO method per se. Based on the consensus estimate directly on the first-order CBO model, we provide a convergence analysis of the first-order CBO method [J. A. Carrillo, S. Jin, L. Li and Y. Zhu, A consensus-based global optimization method for high dimensional machine learning problems, https://arxiv.org/abs/1909.09249v1] without resorting to the corresponding mean-field model. Our convergence analysis consists of two steps. In the first step, we show that the CBO model exhibits a global consensus time asymptotically for any initial data, and in the second step, we provide a sufficient condition on system parameters — which is dimension independent — and initial data which guarantee that the converged consensus state lies in a small neighborhood of the global minimum almost surely. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-09-19T07:00:00Z DOI: 10.1142/S0218202520500463 Issue No:Vol. 30, No. 12 (2020)
Authors:Zhilei Liang, Dehua Wang Pages: 2445 - 2486 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 12, Page 2445-2486, November 2020. A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-10-23T07:00:00Z DOI: 10.1142/S0218202520500475 Issue No:Vol. 30, No. 12 (2020)
Authors:Rafael Bailo, José A. Carrillo, Hideki Murakawa, Markus Schmidtchen Pages: 1 - 36 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We study an implicit finite-volume scheme for nonlinear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced in [R. Bailo, J. A. Carrillo and J. Hu, Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient flow structure, arXiv:1811.11502.]. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-11-12T08:00:00Z DOI: 10.1142/S0218202520500487
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