Global Asymptotic Behavior of Iterative Implicit Schemes

Abstract

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2×2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

“…This includes adaptive temporal and spatial schemes, grid adaptation as an integral part of the numerical solution process, and, most of all, adaptive numerical dissipation controls. Using tools from dynamical systems, Yee et al (1991Yee et al ( -1997, Yee & Sweby (1993-1997, Griffiths et al (1992a,b) and Lafon & Yee (1991 showed that adaptive temporal and adaptive spatial schemes are important in minimizing numerically induced chaos, numerically induced chaotic transients and the false prediction of flow instability by direct numerical simulation (DNS). Their studies further indicate the need in the development of practical adaptive temporal schemes based on error controls to minimize spurious numerics due to the full diseretizations.…”

“…For a combination of initial condition and time step, a super-stable scheme can stabilize unstable physical (analytic) steady states (Yee & Sweby 1993. Super-stable scheme here refers to the region of numerical stability enclosing the physical instability of the true solution of the governing equation.…”

“…Blocks for Reliable Simulations and time and are different from the underlying governing PDE (Yee & Sweby 1993). The procedures of solving the nonlinear algebraic systems resulting from using implicit methods can interfere with or superpose unwanted behavior on the underlying scheme.…”

“…The procedures of solving the nonlinear algebraic systems resulting from using implicit methods can interfere with or superpose unwanted behavior on the underlying scheme. Also, the same scheme can exhibit different spurious behavior when used for time-accurate vs. time marching to the steady states.For a combination of initial condition and time step, a super-stable scheme can stabilize unstable physical (analytic) steady states (Yee & Sweby 1993. Super-stable scheme here refers to the region of numerical stability enclosing the physical instability of the true solution of the governing equation.…”

On uncertainties in simulations in engineering design: A statistical tolerance analysis application

“…Yee and Sweby provide anomalies of implicit approximation procedures. 28 Programming: Mainly unknown mistakes occur which are traced back to human failure. Principally, Hatton's study on syntax errors in commercial software based on language C (basic population: 3.3 million program lines) claims that 8 errors per 1000 lines occur on average.…”