MDPI - Publisher of Open Access Journals

21 pages, 851 KiB

Open AccessArticle

A Finite Difference Method for Solving the Wave Equation with Fractional Damping

by Manruo Cui, Cui-Cui Ji and Weizhong Dai

Math. Comput. Appl. 2024, 29(1), 2; https://doi.org/10.3390/mca29010002 - 29 Dec 2023

Cited by 2 | Viewed by 3595

In this paper, we develop a finite difference method for solving the wave equation with fractional damping in 1D and 2D cases, where the fractional damping is given based on the Caputo fractional derivative. Firstly, based on the weighted method, we propose a [...] Read more.

In this paper, we develop a finite difference method for solving the wave equation with fractional damping in 1D and 2D cases, where the fractional damping is given based on the Caputo fractional derivative. Firstly, based on the weighted method, we propose a new numerical approximation for the Caputo fractional derivative and apply it for the 1D case to obtain a time-stepping method. We then develop an alternating direction implicit (ADI) scheme for the 2D case. Using the discrete energy method, we prove that the proposed difference schemes are unconditionally stable and convergent in both 1D and 2D cases. Finally, several numerical examples are given to verify the theoretical results. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

► Show Figures

Figure 1

18 pages, 774 KiB

Open AccessArticle

Stability and Convergence Analysis of Multi-Symplectic Variational Integrator for Nonlinear Schrödinger Equation

by Siqi Lv, Zhihua Nie and Cuicui Liao

Mathematics 2023, 11(17), 3788; https://doi.org/10.3390/math11173788 - 4 Sep 2023

Cited by 1 | Viewed by 1386

Abstract

Stability and convergence analyses of the multi-symplectic variational integrator for the nonlinear Schr

\ddot{o}

dinger equation are discussed in this paper. The variational integrator is proved to be unconditionally linearly stable using the von Neumann method. A priori error bound for the [...] Read more.

Stability and convergence analyses of the multi-symplectic variational integrator for the nonlinear Schr

\ddot{o}

dinger equation are discussed in this paper. The variational integrator is proved to be unconditionally linearly stable using the von Neumann method. A priori error bound for the scheme is given from the Sobolev inequality and the discrete conservation laws. Subsequently, the variational integrator is derived to converge at

O (Δ x^{2} + Δ t^{2})

in the discrete

L^{2}

norm using the energy method. The numerical experimental results match our theoretical derivation. Full article

(This article belongs to the Section E: Applied Mathematics)

► Show Figures

Figure 1

16 pages, 2034 KiB

Open AccessArticle

Unconditionally Stable System Incorporated Factorization-Splitting Algorithm for Blackout Re-Entry Vehicle

by Yi Wen, Junxiang Wang and Hongbing Xu

Electronics 2023, 12(13), 2892; https://doi.org/10.3390/electronics12132892 - 30 Jun 2023

Cited by 4 | Viewed by 1158

Abstract

A high-temperature plasma sheath is generated on the surface of the re-entry vehicle through the conversion of kinetic energy to thermal and chemical energy across a strong shock wave at the hypersonic speed. Such a condition results in the forming of a blackout [...] Read more.

A high-temperature plasma sheath is generated on the surface of the re-entry vehicle through the conversion of kinetic energy to thermal and chemical energy across a strong shock wave at the hypersonic speed. Such a condition results in the forming of a blackout which significantly affects the communication components. To analyze the re-entry vehicle at the hypersonic speed, an unconditionally stable system incorporated factorization-splitting (SIFS) algorithm is proposed in finite-difference time-domain (FDTD) grids. The proposed algorithm shows advantages in the entire performance by simplifying the update implementation in multi-scale problems. The plasma sheath is analyzed based on the modified auxiliary difference equation (ADE) method according to the integer time step scheme in the unconditionally stable algorithm. Higher order perfectly matched layer (PML) formulation is modified to simulate open region problems. Numerical examples are carried out to demonstrate the performance of the algorithm from the aspects of target characteristics and antenna model. From resultants, it can be concluded that the proposed algorithm shows considerable accuracy, efficiency, and absorption during the simulation. Meanwhile, plasma sheath significantly affects the communication and detection of the re-entry vehicle. Full article

(This article belongs to the Special Issue Advances in Electromagnetic Interference and Protection)

► Show Figures

Figure 1

32 pages, 4095 KiB

Open AccessArticle

Sub-Diffusion Two-Temperature Model and Accurate Numerical Scheme for Heat Conduction Induced by Ultrashort-Pulsed Laser Heating

by Cuicui Ji and Weizhong Dai

Fractal Fract. 2023, 7(4), 319; https://doi.org/10.3390/fractalfract7040319 - 8 Apr 2023

Cited by 2 | Viewed by 1969

Abstract

In this study, we propose a new sub-diffusion two-temperature model and its accurate numerical method by introducing the Knudsen number (

K_{n}

) and two Caputo fractional derivatives (

0 < α, β < 1

) in time into the parabolic [...] Read more.

In this study, we propose a new sub-diffusion two-temperature model and its accurate numerical method by introducing the Knudsen number (

K_{n}

) and two Caputo fractional derivatives (

0 < α, β < 1

) in time into the parabolic two-temperature model of the diffusive type. We prove that the obtained sub-diffusion two-temperature model is well posed. The numerical scheme is obtained based on the

L 1

approximation for the Caputo fractional derivatives and the second-order finite difference for the spatial derivatives. Using the discrete energy method, we prove the numerical scheme to be unconditionally stable and convergent with

O (τ^{min {2 - α, 2 - β}} + h^{2})

, where

τ, h

are time and space steps, respectively. The accuracy and applicability of the present numerical scheme are tested in two examples. Results show that the numerical solutions are accurate, and the present model and its numerical scheme could be used as a tool by changing the values of the Knudsen number and fractional-order derivatives as well as the parameter in the boundary condition for analyzing the heat conduction in porous media, such as porous thin metal films exposed to ultrashort-pulsed lasers, where the energy transports in phonons and electrons may be ultraslow at different rates. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

► Show Figures

Figure 1

28 pages, 17111 KiB

Open AccessArticle

New Insights into a Three-Sub-Step Composite Method and Its Performance on Multibody Systems

by Yi Ji, Huan Zhang and Yufeng Xing

Mathematics 2022, 10(14), 2375; https://doi.org/10.3390/math10142375 - 6 Jul 2022

Cited by 6 | Viewed by 2181

Abstract

This paper develops a new implicit solution procedure for multibody systems based on a three-sub-step composite method, named TTBIF (trapezoidal–trapezoidal backward interpolation formula). The TTBIF is second-order accurate, and the effective stiffness matrices of the first two sub-steps are the same. In this [...] Read more.

This paper develops a new implicit solution procedure for multibody systems based on a three-sub-step composite method, named TTBIF (trapezoidal–trapezoidal backward interpolation formula). The TTBIF is second-order accurate, and the effective stiffness matrices of the first two sub-steps are the same. In this work, the algorithmic parameters of the TTBIF are further optimized to minimize its local truncation error. Theoretical analysis shows that for both undamped and damped systems, this optimized TTBIF is unconditionally stable, controllably dissipative, third-order accurate, and has no overshoots. Additionally, the effective stiffness matrices of all three sub-steps are the same, leading to the effective stiffness matrix being factorized only once in a step for linear systems. Then, the implementation procedure of the present optimized TTBIF for multibody systems is presented, in which the position constraint equation is strictly satisfied. The advantages in accuracy, stability, and energy conservation of the optimized TTBIF are validated by some benchmark multibody dynamic problems. Full article

(This article belongs to the Special Issue Application of Mathematical Method and Models in Dynamic System)

► Show Figures

Figure 1

9 pages, 3269 KiB

Open AccessArticle

A Linear, Second-Order, and Unconditionally Energy-Stable Method for the L²-Gradient Flow-Based Phase-Field Crystal Equation

by Hyun Geun Lee

Mathematics 2022, 10(4), 548; https://doi.org/10.3390/math10040548 - 10 Feb 2022

Cited by 1 | Viewed by 1606

Abstract

To solve the

L^{2}

-gradient flow-based phase-field crystal equation accurately and efficiently, we present a linear, second-order, and unconditionally energy-stable method. We first truncate the quartic function in the Swift–Hohenberg energy functional. We also put the truncated function in the expansive part [...] Read more.

To solve the

L^{2}

-gradient flow-based phase-field crystal equation accurately and efficiently, we present a linear, second-order, and unconditionally energy-stable method. We first truncate the quartic function in the Swift–Hohenberg energy functional. We also put the truncated function in the expansive part of the energy and add an extra term to have a linear convex splitting. Then, we apply the linear convex splitting to both the

L^{2}

-gradient flow and the nonlocal Lagrange multiplier terms and combine it with the second-order SSP-IMEX-RK method. We prove that the proposed method is mass-conservative and unconditionally energy-stable. Numerical experiments including standard tests in the classical

H^{- 1}

-gradient flow-based phase-field crystal equation support that the proposed method is second-order accurate in time, mass conservative, and unconditionally energy-stable. Full article

► Show Figures

Figure 1

11 pages, 8624 KiB

Open AccessArticle

Efficient Fully Discrete Finite-Element Numerical Scheme with Second-Order Temporal Accuracy for the Phase-Field Crystal Model

by Jun Zhang and Xiaofeng Yang

Mathematics 2022, 10(1), 155; https://doi.org/10.3390/math10010155 - 5 Jan 2022

Cited by 5 | Viewed by 1842

Abstract

In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method [...] Read more.

In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method for time marching. It is not only linear and second-order time-accurate, but also unconditionally energy-stable. We prove the unconditional energy stability rigorously and further carry out various numerical examples to demonstrate the stability and the accuracy of the developed scheme numerically. Full article

► Show Figures

Figure 1

12 pages, 1036 KiB

Open AccessArticle

Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces

by Binhu Xia, Yibao Li and Zhong Li

Mathematics 2020, 8(9), 1486; https://doi.org/10.3390/math8091486 - 2 Sep 2020

Cited by 8 | Viewed by 2757

Abstract

This paper describes temporally second-order unconditionally stable direct methods for Allen–Cahn and conservative Allen–Cahn equations on surfaces. The discretization is performed via a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation. We prove that the proposed schemes, which combine a [...] Read more.

This paper describes temporally second-order unconditionally stable direct methods for Allen–Cahn and conservative Allen–Cahn equations on surfaces. The discretization is performed via a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation. We prove that the proposed schemes, which combine a linearly stabilized splitting scheme, are unconditionally energy-stable. The resulting system of discrete equations is linear and is simple to implement. Several numerical experiments are performed to demonstrate the performance of our proposed algorithm. Full article

(This article belongs to the Section E: Applied Mathematics)

► Show Figures

Figure 1

13 pages, 925 KiB

Open AccessFeature PaperArticle

A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional

by Hyun Geun Lee, Jaemin Shin and June-Yub Lee

Mathematics 2019, 7(12), 1242; https://doi.org/10.3390/math7121242 - 16 Dec 2019

Cited by 5 | Viewed by 3206

Abstract

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new [...] Read more.

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme. Full article

► Show Figures

Figure 1

11 pages, 4719 KiB

Open AccessArticle

Application of the Energy-Conserving Integration Method to Hybrid Simulation of a Full-Scale Steel Frame

by Tianlin Pan, Bin Wu, Yongsheng Chen and Guoshan Xu

Algorithms 2016, 9(2), 35; https://doi.org/10.3390/a9020035 - 21 May 2016

Cited by 1 | Viewed by 5559

Abstract

The nonlinear unconditionally stable energy-conserving integration method (ECM) is a new method for solving a continuous equation of motion. To our knowledge, there is still no report about its application on a hybrid test. Aiming to explore its effect on hybrid tests, the [...] Read more.

The nonlinear unconditionally stable energy-conserving integration method (ECM) is a new method for solving a continuous equation of motion. To our knowledge, there is still no report about its application on a hybrid test. Aiming to explore its effect on hybrid tests, the nonlinear beam-column element program is developed for computation. The program contains both the ECM and the average acceleration method (AAM). The comparison of the hybrid test results with thesetwo methods validates the effectiveness of the ECM in the hybrid simulation. We found that the energy error of hybrid test by using ECM is less than that of AAM. In addition, a new iteration strategy with reduction factor is presented to avoid the overshooting phenomena during iteration process with the finite element program. Full article

► Show Figures

Figure 1

Search Results (10)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (10)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI