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We formulate a novel numerical method suitable for the solution of topology optimization problems in solid mechanics. The most salient feature of the new approach is that the space and time discrete equations of the numerical method can be…
A continuum model for the dynamics of a single step with the strongly anisotropic line energy is formulated and analyzed. The step grows by attachment of adatoms from the lower terrace, onto which atoms adsorb from a vapor phase or from a…
This note investigates the stability of both linear and nonlinear switched systems with average dwell time. Two new analysis methods are proposed. Different from existing approaches, the proposed methods take into account the sequence in…
A common paradigm in phase-field models with singular potentials is that global-in-time weak solutions converge to a single equilibrium only after undergoing asymptotic regularization. However, in arXiv:2510.17296 we introduced a novel…
The Cahn-Hilliard equation has been widely employed within various mathematical models in physics, chemistry and engineering. Explicit stabilized time stepping methods can be attractive for time integration of the Cahn-Hilliard equation,…
In this work, we introduce the time filtering technique to develop several innovative semi-discrete schemes in time for the Cahn-Hilliard-Navier-Stokes (CHNS) system. These schemes achieve second-order temporal accuracy while maintaining…
This study introduces the FSSH-2 scheme, a redefined and numerically stable adiabatic Fewest Switches Surface Hopping (FSSH) method for mixed quantum-classical dynamics. It reformulates the standard FSSH hopping probability without…
The main objective of this article is to derive a mathematical theory associated with the nonlinear stability and dynamic transitions of the basic shear flows associated with baroclinic instability, which plays a fundamental role in the…
The development of consistent and stable quasicontinuum models for multi-dimensional crystalline solids remains a challenge. For example, proving stability of the force-based quasicontinuum (QCF) model remains an open problem. In 1D and 2D,…
The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF…
In this paper we focus on the Cahn-Hilliard equation with dynamic boundary conditions, by adding two hyperbolic relaxation terms to the system. We verify that the energy of the total system is decreasing with time. By adding two…
We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditional non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial…
The Cahn-Hilliard equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff an the difficulty to solve it…
We develop an efficient, unconditionally stable, variable step second order exponential time differencing scheme for the incompressible Navier Stokes equations in two and three spatial dimensions under periodic boundary conditions, together…
Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the potential formation of space-filling quasiperiodic structures that lack translational symmetry or decay. In this paper, we…
We study a second order BDF (Backward Differentiation Formula) scheme for the numerical approximation of parabolic HJB (Hamilton-Jacobi-Bellman) equations. The scheme under consideration is implicit, non-monotone, and second order accurate…
Cahn-Hilliard-Navier-Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn-Hilliard-Navier-Stokes system in two and three…
In this paper, we study a second-order accurate and linear numerical scheme for the nonlocal Cahn-Hilliard equation. The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for…
The study of the long time conservation for numerical methods poses interesting and challenging questions from the point of view of geometric integration. In this paper, we analyze the long time energy and magnetic moment conservations of…
In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in $H^{-1}$ and $\mathcal H^s_\alpha$ ($s=1,2,3$) spaces independent of the…