viscous flux treatment in compressible NS sovler
Dear friends, who can tell me how to solve the compressible NavierStokes equations? The governing equations are the 2D compressible viscous NS equations, U_t+F_x+G_y=Fd_x+Gd_y (1) which is written in vector form. U is the vector of conserved variables, F, G are inviscid fluxe vectors and Fd,Gd are viscous flux vectors. I want to apply piecewise parabolic method (PPM) in conjunction with time operator splitting to solve the system of equations. One paper says that the system (1) can reduce to the following three problems, U_t+F_x=0 (2) U_t+G_y=0 (3) U_t=Fd_x+Gd_y (4) (2) and (3) are solved by applying a Godunovtype scheme. (4) is solved via an explicit update and spatial derivative are approximated using central difference. The key problem is I don't know how to solve diffusion equation system (4). The following is my questions: 1. Is there any reference describing the splitting procedure (1)(4)? So far I can't find any one, so I am not sure whether the above splitting procedure is correct. 2. How can the derivatives of viscous flux vectors containing mixed derivative be approximated using central difference approximation? (or how can we solve the diffusion equations)

Re: viscous flux treatment in compressible NS sovl
Dear friends, who can tell me how to solve the compressible NavierStokes equations? The governing equations are the 2D compressible viscous NS equations,
U_t+F_x+G_y=Fd_x+Gd_y (1) which is written in vector form. U is the vector of conserved variables, F, G are inviscid fluxe vectors and Fd,Gd are viscous flux vectors. I want to apply piecewise parabolic method (PPM) in conjunction with time operator splitting to solve the system of equations. One paper says that the system (1) can reduce to the following three problems, U_t+F_x=0 (2) U_t+G_y=0 (3) U_t=Fd_x+Gd_y (4) (2) and (3) are solved by applying a Godunovtype scheme. (4) is solved via an explicit update and spatial derivative are approximated using central difference. The key problem is I don't know how to solve diffusion equation system (4). The following is my questions: 1. Is there any reference describing the splitting procedure (1)(4)? So far I can't find any one, so I am not sure whether the above splitting procedure is correct. 2. How can the derivatives of viscous flux vectors containing mixed derivative be approximated using central difference approximation? (or how can we solve the diffusion equations) 
Re: viscous flux treatment in compressible NS sovl
A good starting point is the last chapter of Anderson Book, once you get the basic programme running, you can introduce the modifications you want. Good Luck

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