Fourier Pseudospectral Method Research Articles

Two energy-preserving Fourier pseudo-spectral methods and error estimate for the Klein–Gordon–Dirac system

We devote the present paper to the convergency of conservative Fourier pseudo-spectral methods for three-dimensional Klein–Gordon–Dirac (KGD) system. By adopting the mathematical induction argument, standard energy method and inverse inequality, the error bound is established under the condition τ<h with time step τ and mesh size h. More specifically, the scalar ϕ and 4-spinor ψ are proved to be convergent in H2 and H1-norm, respectively. In addition, we establish a frame to compute the numerical solutions of three-dimensional KGD system with periodic conditions, and a faster solver is designed to accelerate the computation by means of orthogonal diagonalization technique. Numerical results are reported to verify the error estimate and discrete conservation laws.

Arbitrary high-order linearly implicit energy-conserving schemes for the Rosenau-type equation

In this paper, we present a class of high-order linearly implicit energy-conserving schemes for solving the generalized Rosenau-type equation. We firstly reformulate the equation into a skew-adjoint system with the energy conservation law. Then the nonlinear terms of the system are approximated with an extrapolation/prediction–correction technique and then a linearized energy-conserving system is obtained. Finally, the symplectic Runge–Kutta method in time and the Fourier pseudo-spectral method in space are employed to discretize the resulting linearized system and a fully discrete scheme is obtained, which is linearly implicit, high-order and energy-conserving. Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.

Conformal structure‐preserving schemes for damped‐driven stochastic nonlinear Schrödinger equation with multiplicative noise

AbstractSince many physical phenomena are often influenced by dispersive medium, energy compensation and random perturbation, exploring the dynamic behaviors of the damped‐driven stochastic system has becoming a hot topic in mathematical physics in recent years. In this paper, inspired by the stochastic conformal structure, we investigate the geometric numerical integrators for the damped‐driven stochastic nonlinear Schrödinger equation with multiplicative noise. To preserve the conformal structures of the system, by using symplectic Euler method, implicit midpoint method and Fourier pseudospectral method, we propose three kinds of stochastic conformal schemes satisfying corresponding discrete stochastic multiconformal‐symplectic conservation laws and discrete global/local charge conservation laws. Numerical experiments illustrate the structure‐preserving properties of the proposed schemes, as well as favorable results over traditional nonconformal schemes, which are consistent with our theoretical analysis.

A structure-preserving and variable-step BDF2 Fourier pseudo-spectral method for the two-mode phase field crystal model

For the two-mode phase field crystal models, the evolutions of the solutions and energy vary fast at certain time. To resolve varying time scales efficiently and reduce the computational cost, a variable-step BDF2 Fourier pseudo-spectral method is proposed. It is shown that the fully-discrete scheme is volume-conserving and unconditional energy-stable. Moreover, a robust error estimate is established by using the discrete orthogonal convolution kernels and the corresponding convolution inequalities. Numerical experiments by using the random and adaptive time-stepping strategies are presented to confirm the effectiveness of the scheme.

Improved Uniform Error Bounds on Time-Splitting Methods for the Long-Time Dynamics of the Dirac Equation with Small Potentials

We establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the Dirac equation with small electromagnetic potentials characterized by a dimensionless parameter $\varepsilon\in (0, 1]$ representing the amplitude of the potentials. We begin with a semi-discritization of the Dirac equation in time by a time-splitting method, and then followed by a full-discretization in space by the Fourier pseudospectral method. Employing the unitary flow property of the second-order time-splitting method for the Dirac equation, we prove uniform error bounds at $C(t)\tau^2$ and $C(t)(h^m+\tau^2)$ for the semi-discretization and full-discretization, respectively, for any time $t\in[0,T_\varepsilon]$ with $T_\varepsilon = T/\varepsilon$ for $T > 0$, which are uniformly for $\varepsilon \in (0, 1]$, where $\tau$ is the time step, $h$ is the mesh size, $m\geq 2$ depends on the regularity of the solution, and $C(t) = C_0 + C_1\varepsilon t\le C_0+C_1T$ grows at most linearly with respect to $t$ with $C_0\ge0$ and $C_1>0$ two constants independent of $t$, $h$, $\tau$ and $\varepsilon$. Then by adopting the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the solution and low frequency modes by phase cancellation and energy method, we establish improved uniform error bounds at $O(\varepsilon\tau^2)$ and $O(h^m +\varepsilon\tau^2)$ for the semi-discretization and full-discretization, respectively, up to the long-time $T_\varepsilon$. Numerical results are reported to confirm our error bounds and to demonstrate that they are sharp. Comparisons on the accuracy of different time discretizations for the Dirac equation are also provided.

Rotational Solitary Wave Interactions Over an Obstacle

In this work we investigate the propagation of rotational solitary waves over a submerged obstacle in a vertically sheared shallow water channel with constant vorticity. In the weakly nonlinear regime, the problem is formulated in the forced Korteweg-de Vries framework. The initial value problem for this equation is solved numerically using a Fourier pseudospectral method with integrating factor. Solitary waves are taken as initial data, and the interaction wave-current-topography is analysed. We identify three types of regimes according to the intensity of the vorticity. A rotational solitary wave can bounce back and forth over the obstacle remaining trapped for large times, it can pass over the obstacle without reversing its direction or the wave can be blocked, i.e., it bounces back and forth above the obstacle until reaching a steady state. Such behaviour resembles the classical damped spring-mass system.

Arbitrary high-order exponential integrators conservative schemes for the nonlinear Gross-Pitaevskii equation

In this paper, we propose a family of high-order conservative schemes based on the exponential integrators technique and the symplectic Runge-Kutta method for solving the nonlinear Gross-Pitaevskii equation. By introducing generalized scalar auxiliary variable, the equation is equivalent to a new system with both mass and modified energy conservation laws. Then, a conservative semi-discrete exponential scheme is given by combining the symplectic Runge-Kutta method and the Lawson method in the time direction. Subsequently, we apply the Fourier pseudo-spectral method to approximate semi-discrete system in space and obtain the fully-discrete schemes that conserve the energy and mass. Numerical examples are presented to confirm the accuracy and conservation of the developed schemes.

An improved immersed boundary method by coupling of the multi-direct forcing and Fourier pseudo-spectral methods

The present paper aims to report an improve on the development of the IMERSPEC methodology, which is a methodology that couples the Fourier pseudo-spectral and immersed boundary methods, which can now solve flows over complex geometries. In the present work, the Lagrangian mesh that represents the immersed body inside the flow does not have to coincide with the Eulerian mesh. Furthermore, a high accuracy for the resultant methodology is achieved. These two important improvements arise from the development of the multi-direct forcing method and used to calculate the force field of the immersed boundary method. In order to verify the IMERSPEC methodology, a manufactured solution technique was used. The results of a refined mesh were employed, and different distribution and interpolation functions were used and analysed. Finally, flows over a backward-facing step geometry were simulated in two dimensions. This type of flow presents several physical features, for example, detachment and reattachment of the boundary layer and vortex generation, which are difficult to capture using numerical solutions. A comparison with the experimental data is presented and shows that the IMERSPEC methodology is promising for computational fluid dynamics applications and its analysis.

Numerical analysis of Fourier pseudospectral methods for the Klein–Gordon equation with smooth potentials

Numerical analysis of Fourier pseudospectral methods for the Klein–Gordon equation with smooth potentials

Time-scale investigation with the modified phase field crystal method

This paper extends the two time-scale modified phase-field crystal model to examine crystal plasticity. Two non-linear density dependent functions are constructed to effectively represent hydrostatic strain and dislocation density. The functions are then used to develop a new modified phase field crystal model, which accounts for strain and strain-rate couplings on density dynamics. The non-linear additions provide tunable parameters for controlling dislocation climb versus glide, as well as phonon softening mechanisms. A short-wavelength dampening is also introduced to account for phonon scattering and thermoelastic dissipation processes. Finally, a novel semi-implicit numeric scheme for efficient simulations of this model is presented, which also serves as a generalization of the commonly used Fourier pseudo-spectral method. The technique gives rise to unconditionally stable dynamics with simple numeric implementation requirements.

A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with structure-preservation for long-time dynamics of the Dirac equation with small potentials

A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with structure-preservation for long-time dynamics of the Dirac equation with small potentials

A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation

In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system, which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system. Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem. Under the consistent initial condition, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In addition, the Fourier pseudo-spectral method is used for spatial discretization, resulting in fully discrete energy-preserving schemes. To implement the proposed methods effectively, we present a very efficient iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed algorithms.

Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation

In this paper, a family of high-order linearly implicit exponential integrators conservative schemes is constructed for solving the multi-dimensional nonlinear fractional Schrödinger equation. By virtue of the Lawson transformation and the generalized scalar auxiliary variable approach, the equation is first reformulated to an exponential equivalent system with a modified energy. Then, we construct a semi-discrete conservative scheme by using the Fourier pseudo-spectral method to discretize the exponential system in space direction. After that, linearly implicit energy-preserving schemes which have high accuracy are given by applying the Runge–Kutta method to approximate the semi-discrete system in temporal direction and using the extrapolation method to the nonlinear term. As expected, the constructed schemes can preserve the energy exactly and implement efficiently with a large time step. Numerical examples confirm the constructed schemes have high accuracy, energy-preserving, and effectiveness in long-time simulation.

Spatial resolution of different discretizations over long-time for the Dirac equation with small potentials

We compare the long-time error bounds and spatial resolution of finite difference methods with different spatial discretizations for the Dirac equation with small electromagnetic potentials characterized by ɛ∈(0,1] a dimensionless parameter. We begin with the simple and widely used finite difference time domain (FDTD) methods, and establish rigorous error bounds of them, which are valid up to the time at O(1/ɛ). In the error estimates, we pay particular attention to how the errors depend explicitly on the mesh size h and time step τ as well as the small parameter ɛ. Based on the results, in order to obtain “correct” numerical solutions up to the time at O(1/ɛ), the ɛ-scalability (or meshing strategy requirement) of the FDTD methods should be taken as h=O(ɛ1/2) and τ=O(ɛ1/2). To improve the spatial resolution capacity, we apply the Fourier spectral method to discretize the Dirac equation in space. Error bounds of the resulting finite difference Fourier pseudospectral (FDFP) methods show that they exhibit uniform spatial errors in the long-time regime, which are optimal in space as suggested by the Shannon’s sampling theorem. Extensive numerical results are reported to confirm the error bounds and demonstrate that they are sharp.

A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with energy-preservation for long-time dynamics of the nonlinear Klein-Gordon equation

For the nonlinear Klein-Gordon (NKG) equation with weak nonlinearity characterized by a small parameter ε∈(0,1], the numerical methods for long-time dynamics have received more and more attention. For the NKG equation up to the time at O(1/ε2), the error bounds of classical numerical methods depend significantly on the small parameters ε which creates serious numerical burdens as ε→0+. Recently, an exponential wave integrator Fourier pseudo-spectral (EWIFP) method for the NKG equation has been proposed (Feng et al., 2021) which is uniformly accurate about ε and of course performs well over the classical methods. However, the EWIFP method can not preserve the discrete energy, which is an important structural feature of the true solution in the long-time dynamics from the perspective of geometric numerical integration. In addition, the EWIFP method only is conditionally stable under specific stability condition and the authors did not give a convergence analysis for the method without any CFL condition restrictions on the grid ratio. In this work, we propose an energy-preserving EWIFP (EPEWIFP) method. The proposed method is proved to be time symmetric, unconditionally stable and preserves the discrete energy. We carry out a rigorously error analysis and give uniform error bounds of the method at O(hm0+ε2−βτ2) up to the time at O(1/εβ) with β∈[0,2], mesh size h, time step τ and an integer m0 determined by the regularity conditions. In general, the NKG equation with weak nonlinearity can be converted to an oscillatory NKG equation with wavelength at O(ε2) in time. It is easy to extend the error bounds and energy-preservation properties to the oscillatory NKG equation. Numerical results confirm our error bounds and energy-preservation properties.

Extended Split-Step Fourier Transform Approach for Accurate Characterization of Soliton Propagation in a Lossy Optical Fiber

In this paper, we present a novel extension of the well-known split-step Fourier transform (SSFT) approach for solving the one-dimensional nonlinear Schrödinger equation (NLSE), which incorporates the fiber loss term. While this essential equation governs the pulse propagation in a lossy optical fiber, it is not supported by an exact analytical solution. In this regard, extended versions of the Fourier pseudospectral method (FPSM) and Hopscotch method (HSM) are effectively established as well to cope with the fiber losses effects associated with the pulses’ propagation through the fiber optics, and thus, numerous comparisons are exhaustively conducted among these three compelling numerical approaches to validate their reliability, stability, and accuracy. Based on this, the MATLAB numerical findings bolster that the extended version of the SSFT approach demonstrates superior performance over the other suggested schemes in simulating the solitons propagation in a lossy optical fiber.

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation

In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, the key idea is based on the extrapolation/prediction-correction technique and the symplectic Runge-Kutta method in time, together with the standard Fourier pseudo-spectral method in space. We show that the scheme is linear, high-order, unconditionally stable and preserves the discrete momentum of the system. For the energy-preserving scheme, it is mainly based on the energy quadratization approach and the analogous linearized strategy used in the construction of the linear momentum-preserving scheme. The proposed scheme is linear, high-order and can preserve both the discrete mass and the discrete quadratic energy exactly. Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.

Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs

In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such a novel strategy is based on the newly developed exponential scalar auxiliary variable (ESAV) approach that can remove the bounded-from-below restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products. So it is more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize symplectic Runge-Kutta methods for both solution variables and the auxiliary variable, where the values of the internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms have constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes.

Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

&lt;p style='text-indent:20px;'&gt;In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.&lt;/p&gt;

Second-Order Flows for Computing the Ground States of Rotating Bose-Einstein Condensates

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modelled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. This is judged by the fact that larger size of time steps can be applied in explicit methods, and much smaller number of time steps are needed in both explicit and semi-implicit discretizations, while every time step encounters almost the same computational complexity in comparison with the respective gradient flow type approaches. Rich and detailed numerical examples are documented for verification and comparison.