Modified Coefficients Schemes And It's Application To Rayleigh-Taylor Instability Simulation

In this thesis, we present two new numerical methods named MCupwind andMCENOLLF, respectively, and their applications in the simulation of Rayleigh-Taylor instability encountered in two-phase fluid dynamics. The MCupwind andMCENOLLF are respectively developed from the first-order upwind scheme, and thesecond-order ENO Local Lax-Friedrich scheme without expanding the stencil, butwith a variable coefficient in the respective schemes. Theoretical analysis indicatesthat the MC approach holds the desirable properties which the underlying schemepossessed, and degenerates to the underlying scheme in the badly conditions.The MCupwind schemes are applied in a simulating linear scalar trial model,non-linear evolution trial model (Burgers equation) and shock tube problem, in com-parison with the result of upwind schemes, which has evident improvement exceptthe region including shock. While being applied in a simulating one-dimensionalsteady advection-diffusion problem, the MCupwind schemes produced better re-sults than that of SDFEM on the uniform grid and Shishkin grid, but it is not asgood as that of SDFEM on Bakhvalov grid. Consider the numerical simulation oftwo-dimensional Rayleigh-Taylor instability, we developed a new splitting methodsimilar to the Lax-Friedrichs splitting, then we developed the corresponding MCup-wind scheme. Compared to it's numerical results with those of TVDVL scheme andENOLLF scheme, when coupled with an interface tracking method based on theMAC idea, ghost fluid method and simple adaptive mesh method, the MCupwindscheme has more strong dissipation than the other two schemes. When t=1.5, thenumerical result of MCupwind scheme is not as good as that of TVDVL scheme, butit seems to be better than that of ENOLLF scheme. In addition, the MCupwindscheme uses three nodes for the spatial discretization in one dimension, it is moreeconomical in CPU time than that of the other two schemes. When MCENOLLFscheme is applied in simulating Burgers equation, it improves significantly comparedwith ENOLLF scheme even if near the shock. The numerical results of MCENOLLFscheme has remarkable improvement, compared with that of ENOLLF scheme, whilebeing applied to simulate two-dimensional Rayleigh-Taylor instability.For simulating a high ratio of density (1:100-1:1000) of two-dimensionalRayleigh-Taylor instability, the combined method (interface tracking method, ghostfluid method and adaptive mesh method) yielded better result than that of a singlehigh accurate scheme (such as WENO).