On implicit Godunov’s method with exactly linearized numerical flux

Variation of the exact solution to the Riemann problem caused by a small variation of initial values is studied for compressible Euler equations. By introducing variation matrices associated with the left- and right-hand side states of an initial discontinuity, this variation takes a linear form with regard to variations of initial values, and the exact expressions for these matrices are obtained in a compact explicit form. The variation of the exact Riemann problem solution is used to linearize the numerical flux in the implicit Godunov method. The resulting system of linear algebraic equations written in delta form is then solved in two steps by implementing the Lower-Upper–Symmetric Gauss-Seidel approximate factorization method. An extension of this factorization to unstructured grids is also discussed, which leads to a matrix-free scheme requiring neither matrix storage nor matrix inversion and can be operated with the solution vector and numerical flux only.