DYNAMICAL-APPROACH STUDY OF SPURIOUS STEADY-STATE NUMERICAL-SOLUTIONSOF NONLINEAR DIFFERENTIAL-EQUATIONS .2. GLOBAL ASYMPTOTIC-BEHAVIOR OFTIME DISCRETIZATIONS

The global asymptotic nonlinear behavior of II explicit and implicit t ime discretizations for four 2 x 2 systems of first-order autonomous n onlinear ordinary differential equations (ODEs) is analyzed. The objec tives are to gain a basic understanding of the difference in the dynam ics of numerics between the scalars and systems of nonlinear autonomou s ODEs and to set a baseline global asymptotic solution behavior of th ese schemes for practical computations in computational fluid dynamics . We show how ''numerical'' basins of attraction can complement the bi furcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). W e show how in the presence of spurious asymptotes the basins of the tr ue stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenome non which is not commonly known is that this spurious behavior can res ult in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for f inite time steps. Such distortion, shrinkage and segmentation of the n umerical basins of attraction will occur regardless of the stability o f the spurious asymptotes, and will occur for unconditionally stable i mplicit linear multistep methods. In other words, for the same (common ) steady-state solution the associated basin of attraction of the DE m ight be very different from the discretized counterparts and the numer ical basin of attraction can be very different from numerical method t o numerical method. The results can be used as an explanation for poss ible causes of error, and slow convergence and nonconvergence of stead y-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.