A coupled implicit method for chemical non-equilibrium flows at all speeds

Convergence characteristics of preconditioned Euler equations

“…The preconditioned two-dimensional Euler equations are expressed in the following form [7][8][9][10][11][12][13]19,20]: The matrix C adopted in the present study is the preconditioning matrix of Choi and Merkle [7] represented in the following form:…”

“…In some of the literature [8][9][10]12], the derivative of density with respect to temperature is ignored, leading to a slightly different form for the preconditioning matrix The influence of the derivative of density with respect to temperature will be discussed in detail in Section 2.3.…”

“…Choi and Merkle [7] suggested a preconditioning matrix that introduced well-conditioned eigenvalues and pressure gradient terms. The preconditioning procedure of Choi and Merkle has been extended for use in many CFD applications [8][9][10][11][12][13]. Turkel et al [14] showed that the conventional compressible algorithms without an adequate modulation of the upwinding or equivalently, the artificial viscosity, do not have the correct asymptotic behavior as the Mach number approaches zero.…”

Alleviation of cancellation problem of preconditioned Navier–Stokes equations

“…A number of preconditioners have been suggested to solve the stiffness problem [2][3][4][5][6][7]. Choi and Merkle [5] suggested a preconditioner that introduced well-conditioned eigenvalues and has been extended for use in many CFD applications [6][7][8][9][10][11][12][13].…”

Cancellation problem of preconditioning method at low Mach numbers

“…A number of preconditioning approaches have been suggested to solve the stiffness problem [2][3][4][5][6][7][8][9]. Choi and Merkle [5] suggested a preconditioner that introduced well-conditioned eigenvalues and has been extended for use in many CFD applications [6,7,[10][11][12][13].…”