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In this work, we present a conditionally stable finite-difference scheme that consistently approximates the solution of a general class of (3+1)-dimensional nonlinear equations that generalizes in various ways the quantitative model…
We study the numerical algorithm and error analysis for the Cahn-Hilliard equation with dynamic boundary conditions. A second-order in time, linear and energy stable scheme is proposed, which is an extension of the first-order stabilized…
An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a…
We propose in this paper efficient first/second-order time-stepping schemes for the evolutional Navier-Stokes-Nernst-Planck-Poisson equations. The proposed schemes are constructed using an auxiliary variable reformulation and sophisticated…
In this paper, we present a novel second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities. The scheme combines a standard second order Crank-Nicholson method for the…
The implicit compact finite-difference scheme was developed for evolutionary partial differential parabolic and Schr\"odinger-type equations and systems with a weak nonlinearity. To make a temporal step of the compact implicit scheme we…
The article is dedicated towards the study of fractional order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions…
We present a new temporal discretization paradigm for developing energy-production-rate preserving numerical approximations to thermodynamically consistent partial differential equation systems, called the supplementary variable method. The…
In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that…
Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterised by a conservative part, which can be modelled as a Hamiltonian term, and a…
In this paper, we discuss the time-space Caputo-Riesz fractional diffusion equation with variable coefficients on a finite domain. The finite difference schemes for this equation are provided. We theoretically prove and numerically verify…
We present high-order variational Lagrangian finite element methods for compressible fluids using a discrete energetic variational approach. Our spatial discretization is mass/momentum/energy conserving and entropy stable. Fully implicit…
We develop a fully discrete scheme for time-fractional diffusion equations by using a finite difference method in time and a finite element method in space. The fractional derivatives are used in Caputo sense. Stability and error estimates…
In this paper, by using Strang's second-order splitting method, the numerical procedure for the three-dimensional (3D) space fractional Allen-Cahn equation can be divided into three steps. The first and third steps involve an ordinary…
We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the…
This paper is devoted to studying three-dimensional non-commensurate fractional order differential equation systems with Caputo derivatives. Necessary and sufficient conditions are for the asymptotic stability of such systems are obtained.
We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservation law. The discrete conservation law has the same conserved…
We propose a natural family of higher-order partial differential equations generalizing the second-order Klein-Gordon equation. We characterize the associated model by means of a generalized action for a scalar field, containing…
In this paper, we propose a variational Lagrangian scheme for a modified phase-field model, which can compute the equilibrium states for the original Allen-Cahn type model. Our discretization is based on a prescribed energy-dissipation law…
A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie…