In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson–Nernst–Planck (PNP) equations, which have found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.

We present a regularized finite difference method for the logarithmic Schr ̈odingerequation (LogSE) and establish its error bound. Due to the blowup of the logarithmic nonlinearity, i.e., ln ρ→ −\infty when ρ→0+v with ρ = |u|^2 being the density and u being the complex-valuedwave function or order parameter, there are significant difficulties in designing numerical methodsand establishing their error bounds for the LogSE. In order to suppress the roundoff error and toavoid blowup, a regularized LogSE (RLogSE) is proposed with a small regularization parameter 0 < ε << 1 and linear convergence is established between the solutions of RLogSE and LogSE interm of ε. Then a semi-implicit finite difference method is presented for discretizing the RLogSEand error estimates are established in terms of the mesh sizehand time stepτas well as the smallregularization parameterε. Finally numerical results are reported to illustrate our error bounds.

We present two uniformly accurate numerical methods for discretizing the Zakharovsystem (ZS) with a dimensionless parameter 0< ε ≤ 1, which is inversely proportional to theacoustic speed. In the subsonic limit regime, i.e., 0< ε << 1, the solution of ZS propagates waves with O(ε)- andO(1)-wavelengths in time and space, respectively, and/or rapid outgoing initial layerswith speed O(1/ε) in space due to the singular perturbation of the wave operator in ZS and/or theincompatibility of the initial data. By adopting an asymptotic consistent formulation of ZS, wepresent a time-splitting exponential wave integrator (TS-EWI) method via applying a time-splittingtechnique and an exponential wave integrator for temporal derivatives in the nonlinear Schr ̈odingerequation and wave-type equation, respectively. By introducing a multiscale decomposition of ZS, wepropose a time-splitting multiscale time integrator (TS-MTI) method. Both methods are explicitand convergent exponentially in space for all kinds of initial data, which is uniformly for ε ∈ (0,1].The TS-EWI method is simpler to be implemented and it is only uniformly and optimally accuratein time for well-prepared initial data, while the TS-MTI method is uniformly and optimally accuratein time for any kind of initial data. Extensive numerical results are reported to show their efficiencyand accuracy, especially in the subsonic limit regime. Finally, the TS-MTI method is applied tostudy numerically convergence rates of ZS to its limiting models when ε→0+.

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.