Stabilized Energy Factorization Approach for Allenxe2x80x93Cahn Equation with Logarithmic Floryxe2x80x93Huggins Potential The Allenxe2x80x93Cahn equation is one of fundamental equations of phasefield models, while the logarithmic Floryxe2x80x93Huggins potential is one of the most useful energy potentials in various phasefield models. In this paper, we consider numerical schemes for solving the Allenxe2x80x93Cahn equation with logarithmic Floryxe2x80x93Huggins potential. The main challenge is how to design efficient numerical schemes that preserve the maximum principle and energy dissipation law due to the strong nonlinearity of the energy potential function. We propose a novel energy factorization approach with the stabilization technique, which is called stabilized energy factorization approach, to deal with the Floryxe2x80x93Huggins potential. One advantage of the proposed approach is that all nonlinear terms can be treated semiimplicitly and the resultant numerical scheme is purely linear and easy to implement. Moreover, the discrete maximum principle and unconditional energy stability of the proposed scheme are rigorously proved using the discrete variational principle. Numerical results are presented to demonstrate the stability and effectiveness of the proposed scheme.

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.