Examination of characteristics method with cubic interpolation for advection–diffusion equation

In this study, the use of the characteristics method integrated with the Hermite cubic interpolation or the cubic-spline interpolation on the space line or the time line, i.e., the HCSL scheme, the CSSL scheme, the HCTL scheme, and the CSTL scheme, respectively, for solving the advection–diffusion equation is examined. The advection and diffusion of a Gaussian concentration distribution in a uniform flow with constant diffusion coefficient is used to conduct this investigation. The effects of parameters, such as Peclet number, Courant number, and the reachback number, on these four schemes used herein for solving the advection–diffusion equation are investigated. The simulated results show that the CSSL scheme is comparable to the HCSL scheme, and the two schemes seem insensitive to Courant number as compared with the HCTL scheme and the CSTL scheme. With large Peclet number, for small Courant number the HCTL scheme is more accurate than the HCSL scheme and the CSSL scheme. However, for large Courant number the HCTL scheme has worse computed results in comparison with the HCSL scheme and the CSSL scheme. With small Peclet number, the HCTL scheme, the HCSL scheme, and the CSSL scheme have close simulated results. Despite Peclet number, for small Courant number the CSTL scheme is comparable to the HCTL scheme, but for large Courant number the former scheme provides unacceptable simulated results in which very large numerical diffusion is induced due to the effect of the natural endpoint constraint. For large Peclet number the HCSL scheme and the CSSL scheme integrated with the reachback technique can improve simulated results, but for small Peclet number the HCSL scheme and the CSSL scheme seem not to be influenced by increasing the reachback number.