We propose an explicit numerical method for the periodic Korteweg–de Vries equation. Our method is based on a Lawson-type exponential integrator for time integration and the Rusanov scheme for Burgers’ nonlinearity. We prove first-order convergence in both space and time under a mild Courant–Friedrichs–Lewy condition τ=O(h)⁠, where τ and h represent the time step and mesh size for solutions in the Sobolev space H^3((−π,π))⁠, respectively. Numerical examples illustrating our convergence result are given.

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}).