A high-efficient accurate coupled mesh-free scheme for 2D/3D space-fractional convection-diffusion/Burgers' problems.

This paper first develops a high-efficient accurate mesh-free scheme (CSPH-SFCD) for multi-dimensional time-dependent spatial fractional convection-diffusion (SFCD) equations based on the Riemann-Liouville (RL) derivative in convex domain by coupling a corrected smoothed particle hydrodynamics (CSPH) method with a numerical integral of RL. Subsequently, for the first application, the proposed CSPH-SFCD coupled with up-wind (UW) scheme (CSPH-SFCD-UW) is tentatively extended to investigate the nonlinear shock-wave phenomena in 2D space-fractional Burgers' problem (SFBP) with fractional Neumann boundary at high Reynolds number. The proposed mesh-free scheme for time-dependent SFCD/SFBP with fractional boundary is mainly motived by: (a) the SFCD model with RL fractional calculus is discretized by employing a numerical integral formula and a CSPH scheme, and then a conditionally stable CSPH-SFCD method is first derived; (b) the UW scheme is introduced to stabilize the numerical investigation of convection-dominated SFBP; (c) the fractional Neumann boundary (FNB) condition can be easily enforced in the proposed discretized scheme, and the integer-order NB condition is treated by employing virtual particles interpolation technique; (d) the MPI parallel computing technique is adopted to enhance the computational efficiency. Firstly, the numerical convergence and stability of the proposed scheme are discussed and illustrated by solving several multi-dimensional benchmarks. Secondly, the cases of two-sided and left-sided RL fractional derivatives are also discussed and demonstrated in the numerical tests. The advantages of the proposed method are demonstrated by simulating the 2D/3D SFCD equations in convex domains or under non-uniform point distribution. Finally, the 2D two-component SFBP with FNBC are numerically investigated by using the proposed method, and compared with the FDM results. The complex nonlinear non-local properties governed by SFBP are numerically predicted by the proposed scheme. All the numerical results show that the proposed mesh-free method for time-dependent spatial SFCD problems is high-efficient, flexible and accurate.