scholarly journals Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3050
Author(s):  
Sarita Nandal ◽  
Mahmoud A. Zaky ◽  
Rob H. De Staelen ◽  
Ahmed S. Hendy
Keyword(s):  
Fourth Order ◽  
Numerical Scheme ◽  
Source Term ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Accuracy ◽  
Nonlinear Source ◽  
Second Order In Time ◽  
Convergence Orders ◽  
Temporal Direction

The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2−1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time.

scholarly journals Beyond second-order convergence in simulations of magnetized binary neutron stars with realistic microphysics

2019 ◽  
Vol 490 (3) ◽  
pp. 3588-3600 ◽  
Author(s):  
E R Most ◽  
L Jens Papenfort ◽  
L Rezzolla
Keyword(s):  
Neutron Stars ◽  
Fourth Order ◽  
Finite Temperature ◽  
Equations Of State ◽  
Magnetic Energy ◽  
Second Order ◽  
Conservative Finite Difference Scheme ◽  
Order Scheme ◽  
Convergence Orders ◽  
The Impact

ABSTRACT We investigate the impact of using high-order numerical methods to study the merger of magnetized neutron stars with finite-temperature microphysics and neutrino cooling in full general relativity. By implementing a fourth-order accurate conservative finite-difference scheme we model the inspiral together with the early post-merger and highlight the differences to traditional second-order approaches at the various stages of the simulation. We find that even for finite-temperature equations of state, convergence orders higher than second order can be achieved in the inspiral and post-merger for the gravitational-wave phase. We further demonstrate that the second-order scheme overestimates the amount of proton-rich shock-heated ejecta, which can have an impact on the modelling of the dynamical part of the kilonova emission. Finally, we show that already at low resolution the growth rate of the magnetic energy is consistently resolved by using a fourth-order scheme.

scholarly journals An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Ren ◽  
Lei Liu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Difference Method ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Formula ◽  
Discrete Energy ◽  
Compact Finite Difference ◽  
Second Order In Time

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

scholarly journals On a fractional step-splitting scheme for the Cahn-Hilliard equation

2014 ◽  
Vol 31 (7) ◽  
pp. 1151-1168 ◽  
Author(s):  
A.A. Aderogba ◽  
M. Chapwanya ◽  
J.K. Djoko
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Computational Time ◽  
Fractional Step ◽  
Step Method ◽  
Fractional Step Method ◽  
Order Accuracy ◽  
Content Type ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. Design/methodology/approach – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Findings – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Originality/value – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation.

A Low-Dispersion 3-D Second-Order in Time Fourth-Order in Space FDTD Scheme (<tex>$ M3d_24 $</tex>)

2004 ◽  
Vol 52 (7) ◽  
pp. 1638-1646 ◽  
Author(s):  
H.E.A. El-Raouf ◽  
E.A. El-Diwani ◽  
A.E.-H. Ammar ◽  
F.M. El-Hefnawi
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Second Order In Time ◽  
Low Dispersion

scholarly journals Convergence of a Time Discretization for a Nonlinear Second-Order Inclusion

2017 ◽  
Vol 61 (1) ◽  
pp. 93-120
Author(s):  
Krzysztof Bartosz ◽  
Leszek Gasiński ◽  
Zhenhai Liu ◽  
Paweł Szafraniec
Keyword(s):  
Source Term ◽  
Existence Of Solutions ◽  
Time Discretization ◽  
Second Order ◽  
Clarke Subdifferential ◽  
Smooth Potential ◽  
Locally Lipschitz ◽  
Time Differential ◽  
Second Order In Time ◽  
The Clarke Subdifferential

AbstractWe study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

scholarly journals ON SOME FUČÍK TYPE PROBLEM WITH CUBIC NONLINEARITY ADI APPROACH TO THE PARTICLE DIFFUSION PROBLEM IN MAGNETIC FLUIDS

2011 ◽  
Vol 16 (1) ◽  
pp. 62-71 ◽  
Author(s):  
V. Polevikov ◽  
L. Tobiska
Keyword(s):  
Neumann Boundary ◽  
Magnetic Fluids ◽  
Diffusion Problem ◽  
Second Order ◽  
Particle Diffusion ◽  
Type Problem ◽  
Cubic Nonlinearity ◽  
Order Accuracy ◽  
Second Order In Time ◽  
Ferromagnetic Particles

The present study is devoted to the development of an ADI approach to simulate two-dimensional time-dependent diffusion process of ferromagnetic particles in magnetic fluids. Specific features of the problem are the Neumann boundary conditions. We construct an ADI scheme of formally second order accuracy approximation in time and space. It is proved that the scheme is absolutely stable and it has the accuracy in the energy norm of the second order in time and the order 3/2 in space. The numerical results of a test problem indicate that the convergence rate in space is of the second order as well.

scholarly journals A high-order and fast scheme with variable time steps for the time-fractional Black-Scholes equation

2021 ◽  
Author(s):  
Kerui Song ◽  
Pin Lyu
Keyword(s):  
Fourth Order ◽  
High Order ◽  
Order Accuracy ◽  
Spatial Direction ◽  
Variable Time ◽  
Theoretical Statement ◽  
Black Scholes ◽  
Second Order In Time ◽  
The Stability ◽  
Average Approximation

In this paper, a high-order and fast numerical method is investigated for the time-fractional Black-Scholes equation. In order to deal with the typical weak initial singularities of the solution, we construct a finite difference scheme with variable time steps, where the fractional derivative is approximated by the nonuniform Alikhanov formula and the sum-of-exponentials (SOE) technique. In the spatial direction, an average approximation with fourth-order accuracy is employed. The stability and the convergence with second-order in time and fourth-order in space of the proposed scheme are religiously derived by the energy method. Numerical examples are given to demonstrate the theoretical statement.

scholarly journals Sinc-Galerkin method for solving the time fractional convection–diffusion equation with variable coefficients

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Juan Chen ◽  
MingZhu Li ◽  
Qiang Xu
Keyword(s):  
Diffusion Equation ◽  
Galerkin Method ◽  
Numerical Scheme ◽  
Variable Coefficients ◽  
Exponential Rate ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Exponential Rate Of Convergence ◽  
Time Fractional Derivative

Abstract In this paper, a new numerical algorithm for solving the time fractional convection–diffusion equation with variable coefficients is proposed. The time fractional derivative is estimated using the $L_{1}$ L 1 formula, and the spatial derivative is discretized by the sinc-Galerkin method. The convergence analysis of this method is investigated in detail. The numerical solution is $2-\alpha$ 2 − α order accuracy in time and exponential rate of convergence in space. Finally, some numerical examples are given to show the effectiveness of the numerical scheme.

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