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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.

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In this paper, we present a new methodology to develop arbitrary high-order structure-preserving methods for solving the quantum Zakharov system. The key ingredients of our method are: (i) the original Hamiltonian energy is reformulated into a quadratic form by introducing a new quadratic auxiliary variable; (ii) the original system is then rewritten into a new equivalent system which inherits the quadratic energy based on the energy variational principle; (iii) the resulting system is discretized by symplectic Runge-Kutta method in time combining with the Fourier pseudo-spectral method in space. The proposed method achieves arbitrary high-order accurate in time and can preserve the discrete mass and Hamiltonian energy exactly. Moreover, an efficient iterative solver is presented to solve the resulting discrete nonlinear equations. Finally, extensive numerical examples are provided to demonstrate the theoretical claims and illustrate the efficiency of our method.

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The purpose of this paper is devoted to studying the implicit-explicit (IMEX) one-leg methods for stiff delay differential equations which can be split into the stiff and nonstiff parts. IMEX one-leg methods are composed of implicit one-leg methods for the stiff part and explicit one-leg methods for the nonstiff part. We prove that if the IMEX one-leg methods is consistent of order 2 for the ordinary differential equations, and the implicit one-leg method is A-stable, then the IMEX one-leg methods for stiff delay differential equations are stable and convergent with order 2. Some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the presented methods.

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