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1. Linear, second order and unconditionally energy stable schemes for the viscous Cahn–Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method

   Yang, X; Zhao, J; He, X. (April 2018, Journal of computational and applied mathematics)

   In this paper, we consider numerical approximations for the viscous Cahn–Hilliard equa- tion with hyperbolic relaxation. This type of equations processes energy-dissipative struc- ture. The main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms, allowing large time-marching step, while preserving the energy stability, i.e. the energy dissipative structure at the time-discrete level. We resolve this issue by developing two second-order time-marching schemes using the recently developed ‘‘Invariant Energy Quadratization’’ approach where all nonlinear terms are discretized semi-explicitly. In each time step, one only needs to solve a symmetric positive definite (SPD) linear system. All the proposed schemes are rigorously proven to be unconditionally energy stable, and the second-order convergence in time has been verified by time step refinement tests numerically. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy, and efficiency of the proposed schemes. 

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2. Second order accurate particle-in-cell discretization of the Navier-Stokes equations

   https://doi.org/10.1016/j.jcp.2024.113302  

   Jiang, Han; Schroeder, Craig (December 2024, Journal of Computational Physics)

   We propose the use of PolyPIC transfers [10] to construct a second order accurate discretization of the Navier-Stokes equations within a particle-in-cell framework on MAC grids. We investigate the accuracy of both APIC [16], [17], [8] and quadratic PolyPIC [10] transfers and demonstrate that they are suitable for constructing schemes converging with orders of approximately 1.5 and 2.5 respectively. We combine PolyPIC transfers with BDF-2 time integration and a splitting scheme for pressure and viscosity and demonstrate that the resulting scheme is second order accurate. Prior high order particle-in-cell schemes interpolate accelerations (not velocities) from the grid to particles and rely on moving least squares to transfer particle velocities to the computational grid. The proposed method instead transfers velocities to particles, which avoids the accumulation of noise on particle velocities but requires the polynomial reconstruction to be performed using polynomials that are one degree higher. Since this polynomial reconstruction occurs over the regular grid (rather than irregularly distributed particles), the resulting weighted least squares problem has a fixed sparse structure, can be solved efficiently in closed form, and is independent of particle coverage. 

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3. On the interplay between fluid flow characteristics and small particle deposition in turbulent wall bounded flows

   https://doi.org/10.1063/5.0232440  

   Abbasi, Sanaz; Mehdizadeh, Amirfarhang (November 2024, Physics of Fluids)

   This study investigates the transport and deposition of small particles (500 nm≤dp≤10 μm) in a fully developed turbulent channel flow, focusing on two fluid friction Reynolds numbers: Reτ=180 and Reτ=1000. Using the point particle–direct numerical simulation method under the assumption of one-way coupling, we study how fluid flow (carrier phase) characteristics influence particle deposition. Our findings suggest that changes in flow conditions can significantly alter the deposition behavior of particles with the same size and properties. Furthermore, we show for the first time that gravity has minimal impact on deposition dynamics only at high Reynolds numbers. This research enhances our understanding of small particle deposition and transport in turbulent flows at high Reynolds numbers, which is crucial for various industrial applications. 

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4. Tutorial: Langevin Dynamics methods for aerosol particle trajectory simulations and collision rate constant modeling

   https://doi.org/https://doi.org/10.1016/j.jaerosci.2021.105746  

   Suresh, Vikram; Gopalakrishnan, Ranganathan (February 2021, Journal of aerosol science)

   null (Ed.)

   The Langevin Dynamics (LD) method (also known in the literature as Brownian Dynamics) is routinely used to simulate aerosol particle trajectories for transport rate constant calculations as well as to understand aerosol particle transport in internal and external fluid flows. This tutorial intends to explain the methodological details of setting up a LD simulation of a population of aerosol particles and to deduce rate constants from an ensemble of classical trajectories. We discuss the applicability and limitations of the translational Langevin equation to model the combined stochastic and deterministic motion of particles in fields of force or fluid flow. The drag force and stochastic “diffusion” force terms that appear in the Langevin equation are discussed elaborately, along with a summary of common forces relevant to aerosol systems (electrostatic, gravity, van der Waals, …); a commonly used first order and a fourth order Runge-Kutta time stepping schemes for linear stochastic ordinary differential equations are presented. A MATLAB® implementation of a LD code for simulating particle settling under gravity using the first order scheme is included for illustration. Scaling analysis of aerosol transport processes and the selection of timestep and domain size for trajectory simulations are demonstrated through two specific aerosol processes: particle diffusion charging and coagulation. Fortran® implementations of the first order and fourth order time-stepping schemes are included for simulating the 3D motion of a particle in a periodic domain. Potential applications and caveats to the usage of LD are included as a summary. 

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5. Dynamically Regularized Lagrange Multiplier Method for the Cahn‐Hilliard‐Navier‐Stokes System

   https://doi.org/10.1002/nme.70074  

   Doan, Cao‐Kha; Hoang, Thi‐Thao‐Phuong; Ju, Lili; Lan, Rihui (July 2025, International Journal for Numerical Methods in Engineering)

   ABSTRACT This paper is concerned with efficient and accurate numerical schemes for the Cahn‐Hilliard‐Navier‐Stokes phase field model of binary immiscible fluids. By introducing two Lagrange multipliers for each of the Cahn‐Hilliard and Navier‐Stokes parts, we reformulate the original model problem into an equivalent system that incorporates the energy evolution process. Such a nonlinear, coupled system is then discretized in time using first‐ and second‐order backward differentiation formulas, in which all nonlinear terms are treated explicitly and no extra stabilization term is imposed. The proposed dynamically regularized Lagrange multiplier (DRLM) schemes are mass‐conserving and unconditionally energy‐stable with respect to the original variables. In addition, the schemes are fully decoupled: Each time step involves solving two biharmonic‐type equations and two generalized linear Stokes systems, together with two nonlinear algebraic equations for the Lagrange multipliers. A key feature of the DRLM schemes is the introduction of the regularization parameters which ensure the unique determination of the Lagrange multipliers and mitigate the time step size constraint without affecting the accuracy of the numerical solution, especially when the interfacial width is small. Various numerical experiments are presented to illustrate the accuracy and robustness of the proposed DRLM schemes in terms of convergence, mass conservation, and energy stability. 

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