High-Resolution Algorithms For Tracking Interfaces

Some high-resolution algorithms and applications for tracking interfaces are discussed in this paper. Firstly, central weighted essentially non-oscillatory (CWENO) schemes are briefly described. Secondly, CWENO-type semi-discrete central-upwind schemes are obtained by imposing CWENO reconstructions in semi-discrete central-upwind schemes, and then so-called CWENO-type semi-discrete central-upwind schemes are applied to solve multi-dimensional hyperbolic conservation law(s), convection-diffusion equation, incompressible Euler equations,incompressible Navier-Stokes equations and shallow water equations. Thirdly,triangular semi-discrete central-upwind schemes based on adaptive least squares are proposed to solve multi-dimensional hyperbolic conservation law(s), shallow water equations and Kelvin-Helmholtz instability. Fourthly, to track shock waves effectively in hyperbolic conservation laws in multi-dimensional cases, CWENO schemes are used together with the Level Set method. Fifthly, since CWENO-type semi-discrete central-upwind schemes are combined with the Level Set method and the Ghost Fluid method, the non-reacting shocks problems and detonation discontinuities in multi-material flows are tracked successfully. Finally, a Level Set function is used to detect the object implicitly: the Euler-Lagrange equation is modified to accelerate the convergence rate; simultaneity, the zero contour of the Level Set function would stop on the boundary of the object and detects it.