Article
Mathematics, Applied
D. L. Coelho, E. Vitral, J. Pontes, N. Mangiavacchi
Summary: Computational modeling of pattern formation in nonequilibrium systems is crucial for studying complex phenomena in various scientific fields. The Swift-Hohenberg equation, developed for describing pattern selection near instabilities, has been extended and reviewed in the present paper. The scheme demonstrates strict implementation of various boundary conditions, control parameter distributions, and additional models, showing unconditional stability and second-order accuracy in both time and space.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Ning Cui, Pei Wang, Qi Li
Summary: In this paper, we propose a second-order BDF time marching scheme for Swift-Hohenberg gradient flows, involving quadratic-cubic nonlinearity and vacancy potential. The scheme is constructed based on the multiple scalar auxiliary variables approach and stabilization technique, resulting in an efficient, second-order accurate, and unconditionally energy stable numerical scheme. The uniqueness of solvability and unconditional energy stability are proven, and numerical experiments demonstrate the accuracy and energy stability of the proposed scheme, as well as the effect of quadratic term and vacancy potential on pattern formation.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Optics
Ankita Khanolkar, Yimin Zang, Andy Chong
Summary: The Complex Swift Hohenberg equation (CSHE) has attracted extensive research interest for realistic modeling of mode-locked lasers with saturable absorbers, with various numerical solutions revealing interesting pulse patterns and structures. Experimental demonstration of a CSHE dissipative soliton fiber laser using a unique spectral filter has shown qualitative agreement with numerical simulations, providing insights into dissipative soliton dynamics and potentially opening up new avenues for ultrafast fiber laser research.
PHOTONICS RESEARCH
(2021)
Article
Mathematics
Qingkun Xiao, Hongjun Gao
Summary: This study investigates the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a one-dimensional domain. It demonstrates the occurrence of a stochastic pitchfork bifurcation near critical points and analyzes the case of nearly coinciding bifurcation points.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Hong Sun, Xuan Zhao, Haiyan Cao, Ran Yang, Ming Zhang
Summary: This paper discusses the design, analysis, and numerical simulations of a stabilized variable time-stepping difference scheme for the Swift-Hohenberg equation. The proposed scheme preserves a discrete energy dissipation law and achieves unique solvability and unconditional energy stability through new discrete orthogonal convolution kernels. Additionally, the proposed scheme demonstrates second-order L2 norm convergence in both time and space, independent of the time-step ratios. This is the first time L2 norm convergence of the adaptive BDF2 method is achieved for the Swift-Hohenberg equation.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: In this work, a linear scheme for the Swift-Hohenberg equation based on Leapfrog scheme is introduced, which is energy-stable. The energy dissipation property and second-order accuracy in time of the scheme are rigorously proven. Additionally, a spectral-Galerkin approximation for the spatial variables is adopted, and error estimates for the fully discrete scheme are established. Numerical results are provided to validate the theoretical analysis.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Luigi Amedeo Bianchi, Dirk Bloemker
Summary: The study investigates the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain, using the stochastic Swift-Hohenberg equation with polynomial nonlinearity as an example. By identifying the order where small noise first impacts the bifurcation, the study provides a tool to analyze how the noise influences the dynamics close to a change of stability.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: In this work, a stabilized linear Crank-Nicolson scheme is proposed and analyzed for the Swift-Hohenberg equation. The scheme explicitly treats the nonlinear term and includes two second-order stabilization terms to improve stability. The scheme is shown to satisfy discrete energy dissipation and is rigorously proven to be second-order accurate in time. A spectral-Galerkin approximation is used for the spatial variables, and error estimates are established for the fully discrete scheme. Numerical experiments demonstrate the accuracy and energy stability of the scheme.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Yong-Liang Zhao, Meng Li
Summary: Two full-rank splitting schemes and a low-rank approximation are proposed for the Swift-Hohenberg equation in this paper. Numerical results indicate that these methods are robust, accurate, and energy dissipation-preserving.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: This paper proposes and analyzes a second-order energy stable numerical scheme for the Swift-Hohenberg equation, utilizing mixed finite element approximation in space. The scheme employs a second-order backward differentiation formula scheme with a second-order stabilized term to guarantee long-term energy stability. It is proven to be unconditionally energy stable and uniquely solvable, with an optimal error estimate provided.Numerical experiments are conducted to support the theoretical analysis.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Seunggyu Lee, Sungha Yoon, Junseok Kim
Summary: This paper examines the effective temporal step size for convex splitting schemes in simulating the Swift-Hohenberg equation. It presents specific formulations for different methods and verifies their effectiveness through numerical simulations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Materials Science, Multidisciplinary
Israr Ahmad, Thabet Abdeljawad, Ibrahim Mahariq, Kamal Shah, Nabil Mlaiki, Ghaus Ur Rahman
Summary: This article presents an approximate analytical solution to the fractional order Swift-Hohenberg equation using a novel iterative method called Laplace Adomian decomposition method (LADM). Through this method, nonlinear FSH equations with and without dispersive terms are studied and results are compared through plots for different fractional orders. The article also explores the problem under Caputo-Fabrizio fractional order derivative (CFFOD) and provides examples to demonstrate the results.
RESULTS IN PHYSICS
(2021)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: In this paper, we propose and analyze an unconditionally energy-stable, second-order-in-time, finite element scheme for the Swift-Hohenberg equation. We rigorously prove that our scheme is unconditionally solvable and energy stable. We also demonstrate the boundedness of discrete phase variable for any time and space mesh sizes. Numerical tests are conducted to validate the accuracy and energy stability of our scheme.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: This study introduces a stabilized linear predictor-corrector scheme for the Swift-Hohenberg equation, with a stabilized first-order scheme as the predictor and a stabilized second-order scheme as the corrector. It is rigorously proven that the scheme satisfies the energy dissipation law and is second-order accurate. Numerical experiments demonstrate the accuracy and energy stability of the proposed scheme.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Kevin LI
Summary: The paper examines the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spatial dimension, characterizing the transitions using dynamic transition theory. By applying techniques from center manifold theory, the infinite dimensional problem is reduced to a finite one to analyze possible phase changes depending on the dispersion for every spatial period.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2021)
Article
Peripheral Vascular Disease
Ehsan Rahimi, Soroush Aramideh, Dingding Han, Hector Gomez, Arezoo M. Ardekani
Summary: A mathematical framework for studying subcutaneous drug delivery of mAbs from injection to lymphatic uptake is presented, focusing on tissue biomechanical response and poroelastic properties. The importance of lymph fluid amount at the injection site and injection rate on drug uptake to lymphatic capillaries is highlighted in the results. The study shows that tissue deformability significantly affects tissue poromechanical response and interstitial pressure, impacting short-term lymphatic absorption.
MICROVASCULAR RESEARCH
(2022)
Editorial Material
Computer Science, Interdisciplinary Applications
Jessica Zhang, John Evans, Hector Gomez, Kris van der Zee
ENGINEERING WITH COMPUTERS
(2022)
Article
Materials Science, Multidisciplinary
Michael te Vrugt, Max Philipp Holl, Aron Koch, Raphael Wittkowski, Uwe Thiele
Summary: This paper discusses an active phase field crystal (PFC) model that describes a mixture of active and passive particles. A microscopic derivation from dynamical density functional theory is presented, providing insights into the construction of the nonlinear and coupling terms. The derived model is then investigated using linear stability analysis and nonlinear methods, revealing a rich nonlinear behavior.
MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
(2022)
Article
Multidisciplinary Sciences
C. Henkel, M. H. Essink, T. Hoang, G. J. van Zwieten, E. H. van Brummelen, U. Thiele, J. H. Snoeijer
Summary: The wetting of soft polymer substrates poses several complications compared to rigid substrates. This study presents two models to investigate the wetting of drops in the presence of a strong Shuttleworth effect, revealing the significant role of the Shuttleworth effect in horizontal deformations of the substrate.
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2022)
Article
Mathematics, Applied
A. B. Steinberg, F. Maucher, S. V. Gurevich, U. Thiele
Summary: To analyze pattern formation and phase transitions in Bose-Einstein condensates, an approximate mapping from the nonlocal Gross-Pitaevskii equation to a phase field crystal model is presented. The simplified model allows for exploration of bifurcations and phase transitions through numerical path continuation. The existence of localized states in the PFC approximation is demonstrated, and the impact of higher-order nonlinearities on the bifurcation diagram is discussed.
Article
Chemistry, Multidisciplinary
N. S. Howard, A. J. Archer, D. N. Sibley, D. J. Southee, K. G. U. Wijayantha
Summary: This study demonstrates that the coffee ring effect during colloidal droplet evaporation can be mitigated by adding a specific concentration of a surfactant. Experiments were conducted on carbon nanotube suspensions with different surfactant concentrations on various substrates. The results showed different pattern types, with a critical surfactant concentration resulting in highly uniform deposits with coffee ring subfeatures. Image analysis and profilometry allowed for the identification of these subfeatures and the inference of deposit coverage.
Article
Chemistry, Physical
S. P. Fitzgerald, A. Bailey Hass, G. Diaz Leines, A. J. Archer
Summary: The time evolution of many systems can be modeled by stochastic transitions between energy minima. The availability of time for transitions is crucial when multiple pathways exist. Traditional reaction rate theory is applicable in the long-time limit, but at short times, the system may choose higher energy barriers with higher probability if the distance in phase space is shorter. Simple models are constructed to illustrate this phenomenon, and the geometric minimum action method algorithm is employed to determine the most likely path at both short and long times.
JOURNAL OF CHEMICAL PHYSICS
(2023)
Article
Multidisciplinary Sciences
Jack Paget, Marco G. Mazza, Andrew J. Archer, Tyler N. Shendruk
Summary: Matter self-assembly into layers generates unique properties. We propose a complex tensor order parameter to describe the local degree of lamellar ordering, layer displacement, and orientation. This theory simplifies numerics and facilitates studies on the mesoscopic structure of topologically complex systems.
NATURE COMMUNICATIONS
(2023)
Article
Multidisciplinary Sciences
Tianyi Hu, Hao Wang, Hector Gomez
Summary: We propose a direct van der Waals simulation (DVS) method for computationally studying flows with liquid-vapor phase transformations. Our approach discretizes the Navier-Stokes-Korteweg equations, which couple flow dynamics with van der Waals' theory, enabling simulations of cavitating flows at strongly under-critical conditions and high Reynolds numbers. This technique paves the way for a deeper understanding of phase-transforming flows with applications in science, engineering, and medicine.
Article
Engineering, Multidisciplinary
Saikat Mukherjee, Hector Gomez
Summary: In this paper, a numerical algorithm is proposed to solve the Navier-Stokes-Korteweg equations, which can predict cavitation inception without assumptions about mass transfer. The proposed numerical scheme uses a modified bulk free energy and a Taylor-Galerkin discretization. It is capable of simulating high-speed flows with large pressure gradients, and is demonstrated to be accurate, stable, and robust.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Engineering, Multidisciplinary
Hao Wang, Tianyi Hu, Yu Leng, Mario de Lucio, Hector Gomez
Summary: The MPET2 model combines the multi-network poroelastic theory (MPET) with solute transport equations to predict material deformation, fluid dynamics, and solute transport in porous media. It has wide-ranging applications and a stabilized formulation has been proposed to address numerical discretization challenges.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Physics, Multidisciplinary
Tobias Frohoff-Huelsmann, Uwe Thiele
Summary: This study derives a universal amplitude equation, starting from a linear large-scale oscillatory instability that occurs naturally in many active systems. It can be widely applicable to out-of-equilibrium systems showing spatiotemporal pattern formation. The derived equation belongs to a hierarchical structure for classifying instabilities in homogeneous isotropic systems, combining three features: large-scale vs small-scale instability, stationary vs oscillatory instability, and instability without and with conservation law(s).
PHYSICAL REVIEW LETTERS
(2023)
Article
Physics, Fluids & Plasmas
Andrew J. Archer, Tomonari Dotera, Alastair M. Rucklidge
Summary: This study investigates a class of aperiodic tilings with hexagonal symmetry based on rectangles and two types of equilateral triangles. By designing the pair potentials, aperiodic stable states of rectangle-triangle tilings with two different examples are formed.
Article
Multidisciplinary Sciences
Yu Leng, Pavlos P. Vlachos, Ruben Juanes, Hector Gomez
Summary: In this study, we investigate the collapse and expansion of a cavitation bubble in a deformable porous medium. We propose a continuum-scale model that considers the interaction between compressible fluid flow in the pore network and the elastic response of a solid skeleton. Our findings show that the deformability of the porous medium significantly slows down the collapse and expansion processes, which has important implications for phenomena such as drug delivery and spore dispersion.
Article
Mechanics
Yu Leng, Tianyi Hu, Sthavishtha R. Bhopalam, Hector Gomez
Summary: In this work, we studied the numerical solutions of a gradient-elastic Kirchhoff plate model on convex and concave geometries. For convex plates, we showed the well-posedness of the model and split the higher-order PDE into a system of second-order PDEs. The resulting system was found to match the solution of the original PDE. However, for concave pie-shaped plates, the split method led to significantly different results and even nonphysical solutions. Therefore, the direct method with isogeometric analysis is necessary for gradient-elastic Kirchhoff plates with concave corners.
JOURNAL OF MECHANICS
(2022)
