We establish uniform error estimates of finite difference methods for the nonlinear Schrödinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a ...dimensionless parameter ε (ε Є (0,1). When ε -> 0⁺, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i. e., 0 < ε < 1, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with O(ε²)-wavelength at O(ε⁴) and O(ε²) amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in ε of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in ε, at the order of O(h² + r) and O(h² + r 2/3 ) with time step r and mesh size h for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the l² -norm and discrete semi-H¹ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine ε-dependent error estimates of NLSW, ε-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-H¹ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.
In this study, the generalized finite difference method (GFDM) was used to stably and accurately solve two-dimensional (2D) inverse Cauchy problems in linear elasticity by using the Navier equations. ...In Cauchy problems, overdetermined boundary conditions are imposed on parts of the boundary, whereas there are missing boundary conditions on some parts of the boundary. In Cauchy problems, conventional numerical methods generally generate highly ill-conditioned matrices and thus provide unstable numerical solutions. Moreover, even if a slight noise is added in the boundary conditions, numerical errors are evidently magnified. The GFDM, one of the most promising meshless methods and an extension of the classical finite difference method, can avoid time-consuming tasks of mesh generation and numerical quadrature. The GFDM was applied in this study to stably solve the 2D Cauchy problems in linear elasticity and four numerical examples are provided to illustrate the consistency and accuracy of the presented meshless numerical scheme. Moreover, the stability of the presented scheme for inverse Cauchy problems was proved by adding noise into the boundary conditions.
In the geophysical exploration, the simulation and investigation of wavefields in the porous media including multiple discrete fractures is a challenging problem. The traditional numerical finite ...difference method (FDM) uses extremely fine meshes to describe the thin fractures, leading to tremendous computational costs. In this study, we proposed a modified rotated staggered-grid FDM (RSG FDM) to simulate 3-D wave propagation in the fractured porous media. We derive the equivalent stiffness matrix to replace the fractures by imposing the poroealstic linear-slip interface conditions, thus avoiding discretizing the fracture volumes. For every single time step, the fractures are incorporated into the existing RSG FDM codes by easily modifying the stresses and pressure fields belonging to the fracture zones based on the equivalent medium theory. Through the comparison with the reference solutions of a homogeneous background media with a horizontal fracture, we illustrate the accuracy and efficiency of the proposed method. Furthermore, we present the simulated wavefields for several scenarios including tilted and curved fractures models, multiple finite-length fractures model, and fractured model with complicated free-surface topography. These numerical experiments illustrate the excellent performance of the modified RSG FDM for simulating wavefields in the complicated heterogeneous porous media including multiple fractures, which demonstrates that the proposed method can serve as a reliable and efficient numerical tool for the elastic wave detection of underground porous fractures.
In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell‐Whitehead‐Segel–type equations which are ...very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement.
In this paper, we begin with the nonlinear Schrödinger/Gross–Pitaevskii equation (NLSE/GPE) for modeling Bose–Einstein condensation (BEC) and nonlinear optics as well as other applications, and ...discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, and dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported.
We present the fourth‐order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein–Gordon equation (NKGE), while the nonlinearity strength is characterized ...by ϵp with a constant p ∈ ℕ+ and a dimensionless parameter ϵ ∈ (0, 1. Based on analytical results of the life‐span of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time at O(ϵ−p). We pay particular attention to how error bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ϵ ∈ (0, 1, which indicate that, in order to obtain ‘correct’ numerical solutions up to the time at O(ϵ−p), the ϵ‐scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: h = O(ϵp/4) and τ = O(ϵp/2). It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength at O(1) in space and O(ϵp) in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.
In this study, a new framework for the efficient and accurate solutions of three-dimensional (3D) dynamic coupled thermoelasticity problems is presented. In our computations, the Krylov deferred ...correction (KDC) method, a pseudo-spectral type collocation technique, is introduced to perform the large-scale and long-time temporal simulations. The generalized finite difference method (GFDM), a relatively new meshless method, is then adopted to solve the resulting boundary-value problems. The GFDM uses the Taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables. The method, thus, is truly meshless that can be applied for solving problems merely defined over irregular clouds of points. For problem with complicated geometries, this paper also examines a new distance criterion for adaptive selection of nodes in the GFDM simulations. Preliminary numerical experiments show that the KDC accelerated GFDM methods are very promising for accurate and efficient long-time and large-scale dynamic simulations.
•Apply the GFDM for the first time to 3D dynamic coupled thermoelasticity problems.•Developing a combined KDC-GFDM scheme for long-time dynamic simulations.•Proposing a new distance criterion for the adaptive selection of nodes in the GFDM.•Extremely large time step-sizes can be used in temporal discretization.
•The magnetic fluid heat transfer in biological over a semi-infinite region is investigated.•Absorbing boundary condition is deduced by artificial boundary method.•The superiority of the absorbing ...boundary condition is discussed.•Important parameters of biological heat transfer and magnetohydrodynamic properties are analyzed.
This paper introduces modifications to the classical Pennes’ equation through the Cattaneo correction and considers the influence of a magnetic field on the heat generation of the fluid in a semi-infinite region. The traditional method to treat the problem in unbounded regions is to use a large interval to approximate the infinite region, which leads to large deviations of the solution at the truncated boundaries for a long-time numerical simulation. Using the artificial boundary method, the problem in infinite domains can be converted into an equivalent problem in the bounded domains with the absorbing boundary condition, of which the solution would be more physically reasonable and accurate. The governing equation subject to the absorbing boundary condition is discretized using the finite difference method and leads to effective numerical solutions. We analyze the effects of various parameters including the relaxation parameter, heat transfer coefficient, blood perfusion rate, blood tissue density, magnetic fluid concentration, magnitude, magnetic field frequency, and oscillatory boundary condition on the temperature distribution. The study validates the effectiveness of the absorbing boundary condition when addressing problems in unbounded regions. Some important findings are that the relaxation parameter has a delay effect on magnetic fluid heat transfer and magnetic fluid concentration, and the magnetic field strength and magnetic field frequency are directly proportional to magnetic fluid heat production.
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally ...intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.
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