We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter ε∈(0,1], which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. 0<ε≪1, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with O(ε)-wavelength in time and O(1)-wavelength in space as well as outgoing initial layers in space at speed O(1/ε). This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ. By adapting an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at O(h^2+τ^2/ε) and O(h^2+τ+ε) with h mesh size and τ time step. Thus we obtain a uniform error bound at O(h^2+τ) for 0<ε≤1. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and ε-dependent error bounds between the solutions of KGZ and its limiting model when ε→0+. Finally, numerical results are reported to confirm our error bounds.

We present a uniformly accurate finite difference method and establish rigorously its uniform error bounds for the Zakharov system (ZS) with a dimensionless parameter 0 < ε ≤ 1, which is inversely proportional to the speed of sound. In the subsonic limit regime, i.e., 0 < ε << 1, the solution propagates highly oscillatory waves and/or rapid outgoing initial layers due to the perturbation of the wave operator in the ZS and/or the incompatibility of the initial data, which is characterized by two parameters α ≥ 0 and β ≥ −1. Specifically, the solution propagates waves with wavelength of O(ε) and O(1) in time and space, respectively, and amplitude at O(ε^{min{2,α,1+β}}). This high oscillation of the solution in time brings significant difficulties in designing numerical methods and establishing their error bounds, especially in the subsonic limit regime. A uniformly accurate finite difference method is proposed by reformulating the ZS into an asymptotic consistent formulation and adopting an integral approximation of the oscillatory term. By applying the energy method and using the limiting equation via a nonlinear Schr ̈odinger equation with an oscillatory potential, we rigorously establish two independent error bounds at O(h^2 + τ^2/ε) and O(h^2 + τ^2 + τε^{α^∗} + ε^{1+α^∗}), respectively, with h the mesh size, τ the time step and α^∗ = min{1, α, 1 + β}. Thus we obtain error bounds at O(h^2 + τ^{4/3}) and O(h^2 + τ^{1+α^∗/(2+α^∗)}) for well-prepared (α^∗ = 1) and ill-prepared (0 ≤ α^∗ < 1) initial data, respectively, which are uniform in both space and time for 0 < ε ≤ 1 and optimal at the second order in space. Other techniques in the analysis include the cut-off technique for treating the nonlinearity and inverse estimates to bound the numerical solution. Numerical results are reported to demonstrate our error bounds.

We consider the Schrodinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the Schrodinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations.

We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein–Gordon equation (NKGE) with weak power-type nonlinearity and O(1) initial data, while the nonlinearity strength is characterized by εp with a constant p ∈ N+ and a dimensionless parameter ε ∈ (0, 1], for the long-time dynamics up to the time at O(ε^{−β}) with 0 ≤ β ≤ p. In fact, when 0 < ε ≪ 1, the problem is equivalent to the long-time dynamics of NKGE with small initial data and O(1) nonlinearity strength, while the amplitude of the initial data (and the solution) is at O(ε). By reformulating the NKGE into a relativistic nonlinear Schr ̈odinger equation, we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at O(h^m + ε^{p−β} τ^2) of the TSFP method with h mesh size, τ time step and m ≥ 2 depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at O(ε^{−β}) and uniformly valid for ε ∈ (0, 1]. Especially, the error bounds are uniformly at the second order rate for the large time step τ = O(ε^{−(p−β)/2}) in the parameter regime 0 ≤ β < p. Numerical results are reported to confirm our error bounds in the long-time regime. Finally, the TSFP method and its error bounds are extended to a highly oscillatory complex NKGE which propagates waves with wavelength at O(1) in space and O(ε^β) in time and wave velocity at O(ε^{−β}).

The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications. Due to the singularity of the logarithmic function, it introduces tremendous difficulties in establishing mathematical theories, as well as in designing and analyzing numerical methods for PDEs with such nonlinearity. Here, we take the logarithmic Schrödinger equation (LogSE) as a prototype model. Instead of regularizing f(ρ)=lnρ in the LogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE by first regularizing F(ρ)=ρlnρ−ρ locally near ρ=0+ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrödinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter 0<ε≪1. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improves the linear convergence rate of the regularization method in the literature. Error estimates are also presented for solving the ERLogSE by using Lie–Trotter splitting integrators. Numerical results are reported to confirm our error estimates of the LER and of the time-splitting integrators for the ERLogSE. Finally, our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.

This paper is devoted to developing and analysing a highly accurate conservative method for solving the quantum Zakharov system. The scheme is based on a linearly-implicit compact finite difference discretization and conserve the mass as well as energy in discrete level. Detailed numerical analysis is presented which shows the method is fourth-order accurate in space and second-order accurate in time. Several numerical examples are reported to confirm the conservation properties and high accuracy of the proposed scheme. Finally the compact scheme is applied to study the convergence rate of the quantum Zakharov system to its limiting model in the semi-classical limit.

A Deuflhard-type exponential integrator sine pseudospectral (DEI-SP) method is proposed and analyzed for solving the generalized improved Boussinesq (GIBq) equation. The numerical scheme is based on a second-order exponential integrator for time integration and a sine pseudospectral discretization in space. Rigorous analysis and abundant experiments show that the method converges quadratically and spectrally in time and space, respectively. Finally the DEI-SP method is applied to investigate the complicated and interesting long-time dynamics of the GIBq equation.

We present a uniformly first order accurate numerical method for solving the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters 0 < ε ≤ 1 and 0 <γ ≤ 1, which are inversely proportional to the plasma frequency and the acoustic speed, respectively. In the simultaneous high-plasma-frequency and subsonic limit regime, i.e. ε < γ → 0+, the KGZ system collapses to a cubic Schrödinger equation, and the solution propagates waves with O ( ε2)-wavelength in time and meanwhile contains rapid outgoing initial layers with speed O (1/ γ ) in space due to the incompatibility of the initial data. By presenting a multiscale decomposition of the KGZ system, we propose a multiscale time integrator Fourier pseudospectral method which is explicit, efficient and uniformly accurate for solving the KGZ system for all 0 < ε < γ ≤ 1. Numerical results are reported to show the efficiency and accuracy of scheme. Finally, the method is applied to investigate the convergence rates of the KGZ system to its limiting models when ε < γ → 0+.

Two fundamental difficulties are encountered in the numerical evaluation of time- dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts—a local part that contains the most recent contributions and a history part that contains the con- tributions from all earlier times. The history part is smooth, easily discretized using high-order quadratures, and straightforward to compute using a variety of fast algo- rithms. The local part, however, involves complicated singularities in the underlying Green’s function. Existing methods, based on exchanging the order of integration in space and time, are able to achieve high-order accuracy, but are limited to the case of stationary boundaries. Here, we present a new quadrature method that leaves the order of integration unchanged, making use of a change of variables that converts the singular integrals with respect to time into smooth ones. We have also derived asymptotic formulas for the local part that lead to fast and accurate hybrid schemes, extending earlier work for scalar heat potentials and applicable to moving boundaries. The performance of the overall scheme is demonstrated via numerical examples.

With rheology applications in mind, we present a fast solver for the time-dependent effective viscosity of an infinite lattice containing one or more neutrally buoyant smooth rigid particles per unit cell, in a two-dimensional Stokes fluid with given shear rate. At each time, the mobility problem is reformulated as a 2nd-kind boundary integral equation, then discretized to spectral accuracy by the Nyström method and solved iteratively, giving typically 10 digits of accuracy. Its solution controls the evolution of particle locations and angles in a first-order system of ordinary differential equations. The formulation is placed on a rigorous footing by defining a generalized periodic Green’s function for the skew lattice. Numerically, the periodized integral operator is split into a near image sum— applied in linear time via the fast multipole method—plus a correction field solved cheaply via proxy Stokeslets. We use barycentric quadratures to evaluate particle interactions and velocity fields accurately, even at distances much closer than the node spacing. Using first- order time-stepping we simulate, for example, 25 ellipses per unit cell to 3-digit accuracy on a desktop in 1 hour per shear time. Our examples show equilibration at long times, force chains, and two types of blow-ups (jamming) whose power laws match lubrication theory asymptotics.