Second-order Spatial Derivatives Research Articles

An ultraweak-local discontinuous Galerkin method for nonlinear biharmonic Schrödinger equations

This paper proposes and analyzes a fully discrete scheme for nonlinear biharmonic Schrödinger equations. We first write the single equation into a system of problems with second-order spatial derivatives and then discretize the space variable with an ultraweak discontinuous Galerkin scheme and the time variable with the Crank–Nicolson method. The proposed scheme proves to be computationally more efficient compared to the local discontinuous Galerkin method in terms of the number of equations needed to be solved at each single time step, and it is unconditionally stable without imposing any penalty terms. It also achieves optimal error convergence in L2 norm both in the solution and in the auxiliary variable with general nonlinear terms. We also prove several physically relevant properties of the discrete schemes, such as the conservation of mass and the Hamiltonian for the nonlinear biharmonic Schrödinger equations. Several numerical studies demonstrate and support our theoretical results.

The Modified Guyer-Krumhansl Equations Derived From the Linear Boltzmann Transport Equation

Abstract Phonon hydrodynamics originated from the macroscopic energy and momentum balance equations (called Guyer-Krumhansl equations) proposed by Guyer and Krumhansl by solving the linearized Boltzmann transport equation for studying second sound in the normal-process collision dominated phonon transports in an isotropic nonmetallic crystal with a dispersionless frequency spectrum. However, the low-dimensional dielectric materials and semiconductors are anisotropic, and the different branches in their phonon frequency spectrum usually have different group velocities. For such materials, we derive the macroscopic energy and momentum balance equations from the linear Boltzmann transport equation to describe the phonon hydrodynamic transport, and solve the longstanding debate about whether the energy balance equation contains the second-order spatial derivatives of temperature. Finally, by solving the modified Guyer–Krumhansl equations, we find the minimum and maximum values of the length required by the occurrence of second sound in suspended single-layer graphene with the rectangular shape.

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative method that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

On a stabilization of the Ingard-Myers impedance boundary condition and its time domain implementation

It has been well-known that the Ingard-Myers impedance condition, while simple to apply, is subject to the hydrodynamic Kelvin-Helmholtz-type instability due to its use of a vortex sheet in modeling the flow at the liner boundary. Recently, in the development of a time domain boundary element method for acoustic scattering by treated surfaces, it was found that by neglecting a certain second-order spatial derivative term in the Ingard-Myers formulation, the hydrodynamic instability can be avoided. The present paper aims to provide further analysis of this modified condition, hereby referred to as the Truncated Ingard-Myers Impedance Boundary Condition (TIMIBC). It will be shown, based on the dispersion relations of linear waves, that the instability intrinsic to the Ingard-Myers condition is eliminated in the proposed new formulation. Quantitative assessments on the accuracy of the TIMIBC for scattering of acoustic waves by lined surfaces are carried out, and its effectiveness is demonstrated by a numerical example. It is found that the TIMIBC provides a good approximation to the original Ingard-Myers condition for flows of low to mid subsonic Mach numbers. As such, the proposed TIMIBC can offer a practical solution for overcoming the intrinsic instability associated with the Ingard-Myers condition. Moreover, time domain implementation of the TIMIBC is also discussed and illustrated with a numerical example using a finite difference scheme. In particular, a minimization procedure for finding the poles and coefficients of a broadband multipole expansion for the impedance function is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured.

A Second-Order Scheme for the Generalized Time-Fractional Burgers' Equation

Abstract A second-order numerical scheme is proposed to solve the generalized time-fractional Burgers' equation. The time-fractional derivative is considered in the Caputo sense. First, the quasi-linearization process is used to linearize the time-fractional Burgers' equation, which gives a sequence of linear partial differential equations (PDEs). The Crank–Nicolson scheme is used to discretize the sequence of PDEs in the temporal direction, followed by the central difference formulae for both the first and second-order spatial derivatives. The established error bounds (in the L2− norm) obtained through the meticulous theoretical analysis show that the method is second-order convergent in space and time. The technique is also shown to be conditionally stable. Some numerical experiments are presented to confirm the theoretical results.

A local energy-based discontinuous Galerkin method for fourth-order semilinear wave equations

Abstract This paper proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is unconditionally stable and that it achieves optimal convergence in $L^2$ norm for both the solution and the auxiliary variables without imposing penalty terms. A key part of the proof of the stability and convergence analysis is the special choice of the test function for the auxiliary equation involving the time derivative of the displacement variable, which leads to a linear system for the time evolution of the unknowns. Then we can use standard mathematical techniques in discontinuous Galerkin methods to obtain stability and optimal error estimates. We also obtain energy dissipation and/or conservation of the scheme by choosing simple and mesh-independent interelement fluxes. Several numerical experiments are presented to illustrate and support our theoretical results.

Global existence for a class of non-equilibrium reaction–diffusion systems with flux limitation

For a class of general reaction–diffusion systems with flux limitation in one dimension, when homogeneous Neumann boundary conditions are considered, we will prove global upper and lower boundness estimates for the positive classical solutions and boundness for their first-order spatial derivatives. In particular, the non-equilibrium radiation flux-limited diffusion coupled with material heat conduction model is included as a special case of the general reaction–diffusion systems. The key point of our method is to at first prove that the second-order spatial derivative of the solution of the second diffusion equation can be linearly controlled by the first-order spatial derivative of the solution of the flux-limited diffusion equation. Then, with this control, we can apply the Bernstein method to the flux-limited diffusion equation, while a stratifying time technique is introduced to get a complete first order spatial derivative estimate. With these global prior estimates, the global existence and uniqueness of positive classical solutions for the general flux-limited diffusion systems are proved.

High-order compact difference methods for solving two-dimensional nonlinear wave equations

<abstract><p>Nonlinear wave equations are widely used in many areas of science and engineering. This paper proposes two high-order compact (HOC) difference schemes with convergence orders of $ O\left({{\tau ^4} + h_x^4 + h_y^4} \right) $ that can be used to solve nonlinear wave equations. The first scheme is a nonlinear compact difference scheme with three temporal levels. After calculating the second-order spatial derivatives of the previous time level using the Padé scheme, numerical solutions of the next time level are obtained through repeated iterations. The second scheme is a three-level linearized compact difference scheme. Unlike the first scheme, iterations are not required and it obtains numerical solutions through an explicit calculation. The two proposed schemes are applied to solutions of the coupled sine-Gordon equations. Finally, some numerical experiments are presented to confirm the effectiveness and accuracy of the proposed schemes.</p></abstract>

Compact difference scheme for time-fractional nonlinear fourth-order diffusion equation with time delay

We construct a compact difference scheme for two-dimensional time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the linearized compact difference scheme is formulated by employing the L2−1σ formula and the compact difference operator to approximate temporal Caputo derivative and spatial second-order derivatives, respectively. The unique solvability of the proposed scheme is proved. Then the scheme is proved to be stable and convergent with the rate of second order in time and fourth order in space. The efficiency of the scheme is supported by some numerical experiments.

A second-order accurate Crank–Nicolson finite difference method on uniform meshes for nonlinear partial integro-differential equations with weakly singular kernels

A second-order accurate Crank–Nicolson finite difference method on uniform meshes is proposed and analyzed for a class of nonlinear partial integro-differential equations with weakly singular kernels. The first-order time derivative is approximated by using a Crank–Nicolson time-stepping technique, and the singular integral term is treated by a product averaged integration rule, which preserves the positive semi-definite property of the singular integral operator. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative, and a suitable second-order discretization is adopted for the nonlinear convection term. The solvability, stability and convergence of the method are rigorously proved by the discrete energy method, the positive semi-definite property of the associated quadratic form of the method and a perturbation technique. The error estimation shows that the method has the optimal second-order convergence in time and space for non-smooth solutions. Newton’s iterative method and its algorithm implementation are presented to solve the resulting nonlinear system. Numerical results confirm the theoretical convergence result and show the effectiveness of the method.

A novel numerical viscosity for fourth order hybrid entropy stable shock capturing schemes for convection diffusion equation

This work presents a novel numerical viscosity for constructing fourth-order hybrid entropy stable schemes for convection-diffusion equations. The numerical diffusion is designed using the fourth-order central difference approximation of second-order spatial derivative. By construction, the proposed numerical diffusion is efficient compared to the existing approaches, which depend on higher order jump in the scaled entropy variable using computationally expensive sign stable higher order reconstructions and a suitable diffusion matrix. The resulting schemes are shown entropy stable. The constructed schemes are extended to the systems and further modified using a hybrid approach to suppress the oscillations near discontinuities. Numerical results are presented to support the implementation of the proposed schemes.

On the effects of a beam-like piezoelectric passive controller on the linear stability of the visco-elastic Beck’s beam

The linear stability of the visco-elastic Beck’s beam is investigated in this paper. The mechanical system is equipped with a piezoelectric distributed controller, whose governing equation of motion in linear dynamics resembles that of the beam, in except for the second-order spatial derivative term related to the follower force. Suitable passive control strategies are explored with the aim to increase the Hopf bifurcation critical load, triggered by the presence of the follower force in the mechanical sub-system. The equations of motion of the resultant Piezo-Electro-Mechanical (PEM) system are derived via a variational approach and discretized through the Galerkin method. The linear stability analysis of the PEM system is carried out by solving the descending eigenvalue problem. In this framework, parametric investigations are developed in order to investigate the effects of the electrical parameters on the stability of the system.

Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity

This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank–Nicolson method for the time derivative, and the product-integration (PI) rule is employed to deal with Riemann–Liouville fractional integral term. From which, the non-uniform meshes are utilized to compensate for the singular behavior of the exact solution at t=0 so that our method can reach second-order convergence for time. In order to formulate a fully discrete implicit difference scheme, we employ a standard centered difference formula for the second-order spatial derivative, and the Galerkin method based on piecewise linear test functions is used to approximate the nonlinear convection term. Then we derive the existence and uniqueness of numerical solutions for the proposed implicit difference scheme. Meanwhile, the stability and convergence are proved by means of the energy argument. Furthermore, to demonstrate the effectiveness of the proposed method, we utilize a fixed point iterative algorithm to calculate the discrete scheme. Finally, numerical experiments illustrate the feasibility and efficiency of the proposed scheme, in which numerical results are consistent with our theoretical analysis.

High-order structure-preserving Du Fort–Frankel schemes and their analyses for the nonlinear Schrödinger equation with wave operator

Du Fort–Frankel-type finite difference methods (DFFT-FDMs) are famous for good stability and easy implementation. In this study, by a perfect combination of the classical fourth-order difference to approximate the second-order spatial derivatives with the idea of DFFT-FDMs, a class of high-order structure-preserving DFFT-FDMs (SP-DFFT-FDMs) are firstly developed for solving the periodic initial–boundary value problems (PIBVPs) of one-dimensional (1D) and two-dimensional (2D) nonlinear Schrödinger equations with wave operator (NLSW), respectively. By using the discrete energy method, it is shown that their solutions satisfy the discrete energy- and mass-conservative laws, and are conditionally convergent with an order of O(τ2+hx4+(τhx)2) and O(τ2+hx4+hy4+(τhx)2+(τhy)2) in the discrete H1-norm, respectively. Here, τ denotes time step size, while, hx and hy represent spatial mesh sizes in x and y directions, respectively. Then, by supplementing a stabilized term, a type of stabilized SP-DFFT-FDMs are devised. They not only preserve the discrete energy- and mass-conservation laws, but also own much better stability than original SP-DFFT-FDMs. Finally, numerical results confirm the theoretical findings, and the efficiency of our algorithms.

Spin–phonon dispersion in magnetic materials

Microscopic coupling between the electron spin and the lattice vibration is responsible for an array of exotic properties from morphic effects in simple non magnets to magnetodielectric coupling in multiferroic spinels and hematites. Traditionally, a single spin–phonon coupling constant is used to characterize how effectively the lattice can affect the spin, but it is hardly enough to capture novel electromagnetic behaviors to the full extent. Here, we introduce a concept of spin–phonon dispersion to project the spin moment change along the phonon crystal momentum direction, so the entire spin change can be mapped out. Different from the phonon dispersion, the spin–phonon dispersion has both positive and negative frequency branches even in the equilibrium ground state, which correspond to the spin enhancement and spin reduction, respectively. Our study of bcc Fe and hcp Co reveals that the spin force matrix, that is, the second-order spatial derivative of spin moment, is similar to the vibrational force matrix, but its diagonal elements are smaller than the off-diagonal ones. This leads to the distinctive spin–phonon dispersion. The concept of spin–phonon dispersion expands the traditional Elliott-Yafet theory in nonmagnetic materials to the entire Brillouin zone in magnetic materials, thus opening the door to excited states in systems such as CoF2 and NiO, where a strong spin-lattice coupling is detected in the THz regime.

An optimized finite-difference method to minimize numerical dispersion of acoustic wave propagation using a genetic algorithm

The finite-difference method (FDM) is one of the most popular methods for numerical simulation of wave propagation. The major challenge that we face in solving a partial differential equation for the wave propagation is numerical dispersion. We use an automated and optimized FDM using a genetic algorithm (GA) to optimally compute second-order spatial derivatives. In our method, the explicit finite-difference coefficients are calculated using the GA to minimize the dispersion (phase velocity) for all wavenumbers without using any specific window function. The amplitudes of the pseudospectral window (finite difference [FD] coefficients) are optimized by making the phase velocity close to the analytical solution at each wavenumber and stability close to that of the conventional FDM. Although FD coefficients in this method depend on velocity, grid spacing, and time step, less dispersive solutions can be achieved by computing suitable FD coefficients for varying cases. It takes only a few seconds of additional computation time depending on the order of approximation (e.g., approximately 20 s for the 12th order of approximation in a machine with 120 GB RAM and 2.0 GHz processor). Comparison of the numerical results from our algorithm with those from an existing pseudospectral method, the conventional FDM, and an existing time-space optimized FDM indicates that the proposed method is less dispersive for any order of approximation.

Convergence and stability of meshfree method based on radial basis function for a hyperbolic partial differential equation with piecewise constant arguments

Meshfree method based on radial basis functions is a popular tool used to numerically solve partial differential equations. This paper deals with the convergence and stability of the proposed method for hyperbolic partial differential equations with piecewise constant arguments. Firstly, we apply central difference to the time variable and θ-weighted finite difference to the original equation simultaneously, then the numerical scheme is obtained by using the meshfree method based on radial basis functions to approximate the unknown function and its second-order spatial derivative. Secondly, the convergence is analysed in -norm and the sufficient conditions for asymptotic stability of the numerical solution are acquired under which the analytic solution is asymptotically stable. Finally, our study is supported by several numerical experiments.

Modelling and numerical synchronization of chaotic system with fractional-order operator

Abstract Numerical solution of nonlinear chaotic fractional in space reaction–diffusion system is considered in this paper on a large but finite spatial domain size x ∈ [0, L] for L ≫ 0, x = x(x, y) and t ∈ [0, T]. The classical order chaotic ordinary differential equation is formulated by introducing the second-order spatial fractional derivative with order β ∈ (1, 2]. This second order spatial derivative is modelled by using the definition of the Riesz fractional derivative. The method of approximation combines the Fourier spectral method with the novel exponential time difference schemes. The proposed technique is known to have gained spectral accuracy over finite difference schemes. Applicability and suitability of the suggested methods are tested on Rössler chaotic system of recurring interests in one and two dimensions.

A B-Spline Quasi Interpolation Crank–Nicolson Scheme for Solving the Coupled Burgers Equations with the Caputo–Fabrizio Derivative

In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of the coupled Burgers equations with the Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. The discrete scheme obtained in this way constitutes a system of algebraic equations associated with a bi-pentadiagonal matrix. We show that the proposed scheme is unconditionally stable. Numerical examples are provided to verify the efficiency of the method.

Second-Order Discontinuities for Simple and Double Layers

Simple and double layers first appeared in electrostatics and later found various applications in mathematical physics. In this paper, we present the jump discontinuity conditions for their second-order spatial derivatives.