Related papers: Numerical Energy Dissipation for Time-Fractional P…
The well-known backward difference formulas (BDF) of the third, the fourth and the fifth orders are investigated for time integration of the phase field crystal model. By building up novel discrete gradient structures of the BDF-$\rmk$…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…
The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have…
A kind of spatial fractional diffusion equations in this paper are studied. Firstly, an L1 formula is employed for the spatial discretization of the equations. Then, a second order scheme is derived based on the resulting semi-discrete…
We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete…
In this work, we develop novel structure-preserving numerical schemes for a class of nonlinear Fokker--Planck equations with nonlocal interactions. Such equations can cover many cases of importance, such as porous medium equations with…
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…
In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound…
First-order energy dissipative schemes in time are available in literature for the Poisson-Nernst-Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite…
The numerical simulation of multiphase flows presents several challenges, namely the transport of different phases within de domain and the inclusion of capillary effects. Here, these are approached by enforcing a discrete…
A time-stepping $L1$ scheme for solving a time fractional Fokker-Planck equation of order $\alpha \in (0, 1)$, with a general driving force, is investigated. A stability bound for the semi-discrete solution is obtained for…
In this work, a time-fractional nonlocal diffusion equation is considered. Based on the $L2$-$1_{\sigma}$ scheme on a graded mesh in time and the standard finite element method (FEM) in space, the fully-discrete $L2$-$1_{\sigma}$ finite…
In this paper, we present a systematic framework to derive a Lagrangian scheme for porous medium type generalized diffusion equations by employing a discrete energetic variational approach. Such discrete energetic variational approaches are…
We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on…
We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L 1 initial data, general self-adjoint pure jump L{\'e}vy operators, and locally Lipschitz nonlinearities of…
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…
The shifted fractional trapezoidal rule (SFTR) with a special shift is adopted to construct a finite difference scheme for the time-fractional Allen-Cahn (tFAC) equation. Some essential key properties of the weights of SFTR are explored for…
A new energy and enstrophy conserving scheme is evaluated using a suite of test cases over the global spherical domain or bounded domains. The evaluation is organized around a set of pre-defined properties: accuracy of individual opeartors,…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
In this paper, we derive the time-fractional Cahn-Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze…