Multiple solutions of the transonic potential flow equation

Multiple Steady States for 1-D Transonic Flow

“…We call this phenonemon false stability and demonstrate that it can occur for large time step implicit methods with Courant number as small as 23. As mentioned in the introduction, we believe that analogous numerical phenomena are occurring in the multiple steady state solutions to the steady two-dimensional transonic potential equations in various regimes of free stream Mach number, air foil shape and angle of attack as reported in [9] and [10]. By contrast [13] reports spurious multiple solutions to the nonlinear difference equations which are not related to multiple solutions of a differential equation.…”

“…4.1). This should be contrasted with the high accuracy of the scheme for steady solutions of Burgers' equation (Engquist and Osher [9]) which we regard as somewhat accidental. However, the scheme being monotone, there are no numerical oscillations around the shock.…”

“…To our knowledge, no calculations addressing the issue of (1.7)(2) have been performed, but we strongly suspect, by analogy with much simpler bifurcation problems, that all three solutions are not simultaneously stablenin fact, the symmetric solution is likely to lose its stability in a range of M. However, all three have been numerically computed through a variety of implicit and multigrid methods ( [9]) so there is a real possibility that the phenomenon in (2) of (1.7) also occurs in these calculations. Our main objective here is a detailed investigation of the two questions raised in (1.7) in simpler 1-D transonic model problems.…”

“…Recently, Jameson and Steinhoff [9], [10] have given very convincing numerical evidence that the phenomenon of (1.7)(1) occurs for the potential flow equations by producing three distinct solutions for symmetric Joukowski profiles with zero angle of attack for a range of free stream Mach numbers. To our knowledge, no calculations addressing the issue of (1.7)(2) have been performed, but we strongly suspect, by analogy with much simpler bifurcation problems, that all three solutions are not simultaneously stablenin fact, the symmetric solution is likely to lose its stability in a range of M. However, all three have been numerically computed through a variety of implicit and multigrid methods ( [9]) so there is a real possibility that the phenomenon in (2) of (1.7) also occurs in these calculations.…”

Finite element method in the solution of the Euler and Navier‐Stokes equations for internal flow

“…Multiple solutions to the full p ential equation have stirred quite an interest [6,7]. As a first result of the present multiple solutions to the Euler equations are demonstrated.…”