The Research Of The GRP Scheme For The System Of Inviscid Compressible Fluids

For nearly thirty years, the developed GRP (generalized Riemann problem)schemes as a second order accurate Godunov-type scheme has been widely appliedto the numerical simulation of practical problems, such as the combustion problem,channel fow problem etc. The GRP schemes have two related versions: the originalGRP scheme and the direct Eulerian GRP scheme. The latter one avoids thepassage from the Lagrangian version to the Eulerian version and is easier to extendto the high dimensional problems. This thesis focuses on the research of the GRPscheme for inviscid compressible relativistic and non-relativistic hydrodynamicalEuler systems. The contents are as follows:(1) Developing the second order accurate direct Eulerian GRP schemes for1Dand2D relativistic hydrodynamical (RHD) systems: Because of the appearance ofLorentz factor, the numerical simulations of RHD systems are more difcult andmore complex than the non-relativistical system. Specially, the fuxes and primitivevariables of the RHD systems can not been expressed explicitly by the conservativevariables. So, in order to get the primitive variables, one has to numerically solvea nonlinear algebraic equation, and then obtain the fuxes by the solved primitive variables. During developing of these schemes, we mainly focus on the case thatthe left (right) nonlinear wave in the local GRP is located on the left (right) sideof the cell interface: with the help of the Riemann invariants and shock relations,we can set up the linear system of the limiting values of the total derivatives ofthe velocity and pressure after resolving both nonlinear waves. The coefcientmatrix and the right hand side of this linear system are made of the intermediatestates of the associated Riemann problem and the left and right side states ofthe initial data of the local GRP. Here and below, the limiting values are alongthe cell interface at the place of initial discontinuity of the GRP. After gettingthose total derivatives, we can obtain the limiting values of the frst order timederivatives of the primitive variables, depending on the position of the contactdiscontinuity. For other cases, e.g. nonlinear waves located on the same side ofthe cell interface, transonic rarefaction wave, acoustic wave, the resolution of theGRP can be simplifed to some extent. Take the transonic rarefaction wave case asan example. Because the cell interface is inside the rarefaction wave, we only needto resolve the transonic rarefaction wave by directly using corresponding Riemanninvariants and the defnition of the sonic point. Although the2D GRP schemeis similar to the1D GRP scheme, its deduction is diferent from the1D scheme.This is because the Riemann invariants (except entropy), which correspond to thenonlinear waves of the split system of high dimensional RHD equations, depend onthe tangential velocity nonlinearly, so that the1D scheme can not directly extendto the2D case. It is diferent from the non-relativistical system. Some numericalexperiments are given to demonstrate the accuracy and efectiveness in capturingthe shock and other discontinuities.(2) Developing the third order accurate direct Eulerian GRP schemes for1Dnon-relativistical Euler system and RHD system: Now, we need not only the lim-iting values of the frst order time derivatives of the primitive variables, but alsothe limiting values of their second order time derivatives. Compared with the cal-culation of the frst order time derivatives, the second order time derivatives aregiven through resolving the local GRP more delicately. For example, when the left (resp. right) nonlinear wave is located on the left (right) side of the cell inter-face, we need to calculate the second and third order derivatives of the Riemanninvariants, or derive the relations across the shock trajectory satisfed by the secondorder directional derivatives (along the shock trajectory) of the physical quantities.The limiting values of the second order time derivatives for the transonic case istoo difcult to resolve, and the work of this part is still in consideration. In thecontrast to the non-relativistical Euler system, besides the complexity accompaniedby the highly nonlinear equations, even though for the ideal gas, we can not getthe explicit expressions of the integrations arising from the resolution of the localGRP, and have to use numerical quatrature. Several numerical experiments showthat the schemes can achieve the third order convergent rate and are efective.