Two novel classes of linear highorder structurepreserving schemes for the generalized nonlinear Schrxc3xb6dinger equation Abstract   In this letter, we present two novel classes of linear highorder masspreserving schemes for the generalized nonlinear Schrodinger equation. The original model is first equivalently transformed into a pair of realvalued equations, which are then linearized by the extrapolation technique. We employ the symplectic Rungexe2x80x93Kutta method for the resulting linearized model to derive a class of linear massconserving schemes. To improve the accuracy of the schemes, a predictionxe2x80x93correction strategy is applied to develop a class of predictionxe2x80x93correction schemes. The proposed methods are shown to be linear, masspreserving and may reach high order. To match the high precision of temporal discretization, the Fourier pseudospectral method is utilized for spatial discretization. Numerical results are shown to verify the accuracy and conservation property of the proposed schemes.

PositivityPreserving Finite Difference WENO Scheme for TenMoment Equations with Source Term We develop a positivitypreserving finite difference WENO scheme for the TenMoment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability preserving Rungexe2x80x93Kutta scheme. The positivity of the forward Euler scheme is obtained under a CFL condition by using a scaling type limiter, while the solution of the source operator is performed exactly and is positive without any restriction on the time step. The proposed method can be used with any WENO reconstruction scheme and we demonstrate it with fifth order accurate WENOJS, WENOZ and WENOAO schemes. An adaptive CFL strategy is developed which can be more efficient than the use of reduced CFL for positivity preservation. Numerical results show that high order accuracy and positivity preservation are achieved on a range of test problems.