Cutoff Function Research Articles

Error estimate of the conservative difference scheme for the derivative nonlinear Schrödinger equation

In this letter, we propose and rigorously analyze a fully implicit difference scheme for the derivative nonlinear Schrödinger equation. We show that the numerical scheme at least preserves two discrete conserved quantities. Next, to facilitate error estimate, the numerical scheme is converted into an equivalent system, which can be regarded as one-stage Gaussian–Legendre Runge–Kutta method in time. Furthermore, with the help of the cut-off function technique, we prove the convergence of the equivalent system for the first time with the convergence order O(τ2+h2) under discrete L∞-norm without any restriction on step ratio. Finally, the numerical results confirm theoretical findings and capacity in long-time simulations.

A second order difference method combined with time two-grid algorithm for two-dimensional time-fractional Fisher equation

In this paper, a finite difference (FD) scheme based on time two-grid algorithm is proposed for solving the two-dimensional time-fractional Fisher equation (2D-TFFE). Firstly, the Caputo fractional derivative and the spatial derivative are discretized by the L 2 − 1 σ formula and the central difference formula, respectively. In order to improve the efficiency of computation, the time two-grid algorithm is then constructed. Secondly, based on the cut-off function technic, stability and convergence of the time two-grid FD scheme are obtained by the energy method, and the global convergence order is O ( τ F 2 + τ C 4 + h x 2 + h y 2 ) , where τ F and τ C represent time-step sizes on the fine and coarse grid, respectively, while h x and h y represent the space-step sizes. Finally, numerical experiments are presented to show the feasibility and efficiency of the algorithm.

Sixth-order perturbed WENO interpolation-based AWENO and WCNS-E schemes for hyperbolic conservation laws

The weighted essentially non-oscillatory (WENO) interpolation-based schemes (the alternative WENO (AWENO) scheme and the explicit weighted compact nonlinear (WCNS-E) scheme) are limited to the fifth-order despite the sixth-order Taylor expansion of the numerical flux used. This order reduction is due to the fifth-order accuracy inherent in the WENO interpolation. We investigate the perturbed WENO interpolation with a free parameter and affine-invariant WENO weights to recover sixth-order accuracy in smooth regions. A cutoff function of the free parameter with a threshold, determined by approximate dispersion relation analysis, is applied to enhance the ENO property. The proposed schemes perform better in accuracy, dissipation, resolution, shock-capturing, and efficiency in 1D and 2D benchmark problems.

On the wavefunction cutoff factors of atomic hydrogen confined by an impenetrable spherical cavity

AbstractThe Schrödinger equation for the hydrogen atom enclosed by an impenetrable spherical cavity is solved through a Finite‐Differences approach to gain an insight on the actual nature and structure of the ansatz wavefunction cutoff factor widely used in an ad hoc manner in corresponding variational calculations to comply with the Dirichlet boundary conditions. The results of this work provide a theoretical foundation for the choice of the appropriate analytical cutoff functions that fulfill the boundary conditions. We find three different regions for the behavior of the cutoff functions. Small cavity radius where the cutoff function has a parabolic behavior, an intermediate region where the cutoff function is quasi‐linear, and a large cavity region where the cutoff function is a step‐like function. We deduce the traditional linear and quadratic cutoff functions used in the literature as well as its validity region for the confining radius. Finally, we provide a mathematical deduction of the exact cutoff function in terms of the nodal structure of the free hydrogenic wavefunctions and a relation to the Laguerre polynomials for some cavity radii where the free atomic energy level coincides with a confined energy level. We find that the cutoff function transit over several unconfined solutions in terms of its nodal structure as the system is compressed.

Sequences of nodal solutions for critical double phase problems with variable exponents

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }$$\\end{document} and in the Musielak–Orlicz Sobolev space.

Error estimates of a space–time Legendre spectral method for solving the Korteweg–de Vries equation

In this paper, a space–time spectral method for solving the Korteweg–de Vries equation is considered. The discrete schemes of the method are based on the Legendre–Petrov–Galerkin method in spatial direction and the Legendre-tau method in temporal direction with nonperiodic boundary conditions. Stability analysis results and error estimates are obtained in L2-norm by introducing a cut-off function without Lipschitz condition. The method is also applicable to solve some (2m+1)th-order differential equations. Comparison of our numerical results with those of other spectral methods exhibits the accuracy of our methods for the Korteweg–de Vries equation.

Criticality in the fracture of silica glass: Insights from molecular mechanics.

The universality of avalanches characterizing the inelastic response of disordered materials has the potential to bridge the gap from micro to macroscale. In this study, we explore the statistics and the scaling behavior of avalanches occurring during the fracture process in silica glass using molecular mechanics. We introduce a robust method for capturing and quantifying these avalanches, allowing us to perform rigorous statistical analyses, revealing universal power laws associated with critical phenomena. The influence of an initial crack is explored, observing deviations from mean-field predictions while maintaining the property of criticality. However, the avalanche exponents in the unnotched samples are predicted correctly by the mean-field depinning model. Furthermore, we investigate the strain-dependent probability density function, its cutoff function, and the interrelation between the critical exponents. Finally, we unveil distinct scaling behavior for small and large avalanches of the crack growth, shedding light on the underlying fracture mechanisms in silica glass.

A high-efficiency adaptive TENO scheme with optimal accuracy order for compressible flow simulation

In this paper, we propose a new adaptive cut-off function and develop a fifth-order targeted ENO scheme that achieves optimal accuracy at any order of critical points, which performs excellently in conventional compressible gas dynamics. The cut-off value CT is crucial for controlling the dissipation in TENO schemes. Adjusting CT can improve shock capture and increase dissipation for small-scale features. To address this issue, Fu et al. [27] and Peng et al. [28] developed the TENO-A and TENO-LAD schemes, respectively, with adaptive cut-off values CT. However, these adaptive cutoffs CT are computationally expensive due to additional free parameters and complexity. Furthermore, all TENO schemes, including the adaptive versions, can suffer from loss of accuracy at higher-order critical points. In this paper, a new adaptive cut-off function is proposed that uses a global smoothness indicator correction so that it can be automatically tuned between 0 and a given positive constant λd, ensuring that the new TENO scheme achieves optimal accuracy in smooth regions and good shock-capturing in the less smooth parts. A series of benchmark simulations demonstrate that the presented scheme provides optimal accuracy, better robustness, resolution, and numerical stability, and higher resolution than existing TENO schemes while improving computational efficiency. Due to the simplicity of the new cut-off structure, it can be easily extended to improve other TENO schemes.Program files associated to this article may be found at https://github.com/fourhairs/cfd.

A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers’ equation

This study explores a third-order backward differentiation formula (BDF3) and nonlinear fourth-order difference method (FODM) for solving the generalized viscous Burgers’ equation (GVBE). The BDF3 method is employed for discretizing the time derivative, while the nonlinear term uλux is handled using a newly constructed nonlinear fourth-order difference operator. The spatial second derivative is discretized using a linear fourth-order difference formula. We establish the convergence of the proposed BDF3-FODM by introducing a cut-off function (COF). The main contributions include the construction of a novel nonlinear fourth-order difference operator, addressing the limitation observed in Zhang et al. (2021), Guo et al. (2023), where the order of spatial convergence was restricted to second order. Finally, a numerical test is presented to validate our theoretical analysis.

Global existence and decay estimates of solutions for the compressible Prandtl type equations with small analytic data

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G=defu+y2〈t〉φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ, which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data.

Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation

<abstract><p>This work focuses on exploring pointwise error estimate of three-level conservative difference scheme for supergeneralized viscous Burgers' equation. The cut-off function method plays an important role in constructing difference scheme and presenting numerical analysis. We study the conservative invariant of proposed method, which is energy-preserving for all positive integers $ p $ and $ q $. Meanwhile, one could apply the discrete energy argument to the rigorous proof that the three-level scheme has unique solution combining the mathematical induction. In addition, we prove the $ L_2 $-norm and $ L_{\infty} $-norm convergence of proposed scheme in pointwise sense with separate and different ways, which is different from previous work in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. Numerical results verify the theoretical conclusions.</p></abstract>

Nonlinear dynamical analysis for globally modified incompressible non-Newtonian fluids

We present the global modification of the Ladyzhenskaya equations, for incompressible non-Newtonian fluids. This modification is through a cut-off function that multiplies the convective term of the equation and an additional artificial smoothing dissipation term as part of the viscous term of the equation. The goal of this work is the comparative analysis between the modified system and the non-modified system. Therefore, we show the existence and regularity of weak solutions, the existence of global attractors, the estimation of the fractal dimension of the global attractors, and finally, the relationship of the autonomous dynamics between the modified system and the non-modified system.

Sign changing solutions for critical double phase problems with variable exponent

In this paper, we deal with a double phase problem with variable exponent and a right-hand side consisting of a Carathéodory perturbation defined only locally and of a critical term. We stress that the presence of the critical term inhibits the possibility to apply results of the critical point theory to the corresponding energy functional. Instead, we use suitable cut-off functions and truncation techniques in order to work with a coercive functional. Then, using variational tools and an appropriate auxiliary coercive problem, we can produce a sequence of sign changing solutions to our main problem converging to 0 in L^{\infty} and in the Musielakk–Orlicz Sobolev space.

A magnetic skyrmion diode based on potential well inducting effect

Magnetic skyrmions have great potential in the application of spintronic devices due to their stable topologically protected spin configuration. To meet the needs of spintronic device design, it is necessary to manipulate the movement of the magnetic skyrmions. Here we propose a skyrmion diode based on potential well induced skyrmion motion through theoretical calculations. The potential well is generated by the voltage-controlled magnetic anisotropy (VCMA) gradient. By utilizing the induction of the potential well as well as the skyrmion Hall effect (SkHE), the velocity and trajectory of the skyrmions can be controlled and the forward pass and reverse cutoff functions of diode-like devices have been realized. Furthermore, we report the dynamics of current-driven skyrmions in a racetrack with locally applied VCMA. Under the influence of the SkHE, the difference in dynamic behavior between forward and reverse motion of the skyrmions is obvious, and the potential well can produce different pinning, depinning and annihilating effects on forward and reverse moving skyrmions. Our results can be beneficial for the design and development of magnetic skyrmion diodes.

An elliptic problem of the Prandtl–Batchelor type with a singularity

We establish the existence of at least two solutions of the Prandtl–Batchelor like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondifferentiable, and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequence of C^{1} functionals and a cutoff function. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.

Uniqueness, symmetry and convergence of positive ground state solutions of the Choquard type equation on a ball

This paper is concerned with the qualitative properties of the positive ground state solutions to the nonlocal Choquard type equation on a ball BR. First, we prove the radial symmetry of all the positive ground state solutions by using Talenti's inequality. Next we develop the Newton's Theorem and then resort to the contraction mapping principle to establish the uniqueness of the positive ground state solution. Finally, by constructing cut-off functions and applying energy comparison method, we show the convergence of the unique positive ground state solutions as the radius R→∞. Our results extend and complement the existing ones in the literature.

Collision-Free Adaptive Constrained Tracking of Satellite Clusters Under Time-Receding Horizons

The satellite cluster flying (SCF) technology is essential in contemporary space engineering, while escalating requirements and clustering scales pose challenges for control implementation. This paper provides a low-complexity adaptive control scheme for asymptotic tracking of the SCF system, aiming to drive satellites to form a scalable formation lattice (F-lattice) around the virtual global center (VGC). The topological relations denoted by graphic matrices are replaced with a set of well-defined bump functions, which are also utilized to generate finite cut-off potential functions for preventing collisions. Then, a set of time-receding horizons (TRH) stitched with quadratic Lyapunov functions (QLF) are presented for constrained dynamic performance, relaxing the initial dependencies and specifying the practically preset finite-time stability (PPFS) of the perturbed SCF throughout. In addition, by integrating specific inequality and adaptive estimators, disturbances and recursive derivatives are compensated online without “complexity explosion.” At last, simulations built upon a satellite cluster illustrate the effectiveness and superiority of the proposed method.

Liouville theorems for semilinear differential inequalities on sub-Riemannian manifolds

In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of “good” cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.

Error Estimates of High-order Conservative Schemes for the Generalized Nonlinear Schrödinger Equation in One Dimension

In this paper, we use the high-order difference operator Dμ to compensate the central difference operator . Then, we discretize time by the time-midpoint method. Fully discrete schemes are obtained for the generalized nonlinear Schrödinger equation. For the high-order methods, conservation of mass and energy can be proved theoretically in the discrete sense. Then, to establish the optimal error estimates without any requirement on the grid ratio, we adopt the cut-off function technique. The error estimates of the schemes are proved to be of O(τ2 + h2(μ+1)) with time step size τ and mesh size h. Some numerical experiments are given to verify the accuracy order. In addition, the dynamics of the generalized nonlinear Schrödinger equation in one dimension are simulated.

A linear decoupled physical-property-preserving difference method for fractional-order generalized Zakharov system

In this paper, an efficient physical-property-preserving algorithm for the space fractional-order generalized Zakharov system is proposed and analyzed. Firstly, the space fractional-order generalized Zakharov system is reformulated as an equivalent system of equations by introducing the auxiliary equation. Then the spatial fourth-order physical-property-preserving linearly implicit difference scheme is developed for the transformed system. Subsequently, with the aid of the cut-off function and discrete energy analysis method, the underlying scheme are proven to be with the optimal order of O(Δt2+h4) in discrete L∞ and L2 norms. The main feature of the new scheme is physical-property-preserving, linearly decoupled and easy to be applied in parallel computing, especially in long time simulations and large-scale problems. At last, ample numerical results are exhibited to substantiate the efficiency and preservation properties of our scheme, and investigate the dynamic behaviors of the collision of different solitary waves with subsonic and supersonic propagation speeds, respectively.