A Hybrid ESA-CCD Method for Variable-Order Time-Fractional Diffusion Equations

In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.

Multiderivative Combined Dissipative Compact Scheme Satisfying Geometric Conservation Law III: Characteristic-Wise Hybrid Method

Based on newly developed weight-based smoothness detectors and non-linear interpolations designed to capture discontinuities for the multiderivative com-bined dissipative compact scheme(MDCS),hybrid linear and nonlinear interpolations are proposed to form hybrid MDCS.These detectors are derived from the weights used for the nonlinear interpolations and can provide suitable switches between the linear and the nonlinear schemes to realize the characteristics for the hybrid MDCS of capturing discontinuities and maintaining high resolution in the region without large discontinuities.To save computational cost,the nonlinear scheme with characteris-tic decomposition is only applied in the detected discontinuities region by specially designed hybrid strategy.Typical tests show that the hybrid MDCS is capable of cap-turing discontinuities and maintaining high resolution power for the smooth region at the same time.With the satisfaction of the geometric conservative law(GCL),the MDCS is further applied on curvilinear mesh to present its promising capability of handling pragmatic simulations.

Numerical Simulation of Free Surface by an Area-Preserving Level Set Method

We apply in this study an area preserving level set method to simulate gas/water interface flow.For the sake of accuracy,the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the fifthorder accurate upwinding combined compact difference(UCCD)scheme.This scheme development employs two coupled equations to calculate the first-and second-order derivative terms in the momentum equations.For accurately predicting the level set value,the interface tracking scheme is also developed to minimize phase error of the first-order derivative term shown in the pure advection equation.For the purpose of retaining the long-term accurate Hamiltonian in the advection equation for the level set function,the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme.Also,to keep as a distance function for ensuring the front having a finite thickness for all time,the re-initialization equation is used.For the verification of the optimized UCCD scheme for the pure advection equation,two benchmark problems have been chosen to investigate in this study.The level set method with excellent area conservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam-break,Rayleigh-Taylor instability,two-bubble rising in water,and droplet falling problems.