Lun-Shin Yao

Lun-Shin Yao

Von Neumann established that discretized algebraic equations must be consistent with the differential equations, and must be stable in order to obtain convergent numerical solutions for the given differential equations. The "stability" is required to satisfactorily approximate a differential derivative by its discretized form, such as a finite-difference scheme, in order to compute in computers. His criterion is the necessary and sufficient condition only for steady or equilibrium problems. It is also a necessary condition, but not a sufficient condition for unsteady transient problems; additional care is required to ensure the accuracy of unsteady solutions.

Lun-Shin Yao

Discrete numerical methods with finite time-steps represent a practical technique to solve initial-value problems involving nonlinear differential equations. These methods seem particularly useful to the study of chaos since no analytical chaotic solution is currently available. Using the well-known Lorenz equations as an example, it is demonstrated that numerically computed results and their associated statistical properties are time-step dependent. There are two reasons for this behavior. First, chaotic differential equations are unstable so that any small error is amplified exponentially near an unstable manifold. The more serious and lesser-known reason is that stable and unstable manifolds of singular points associated with differential equations can form virtual separatrices. The existence of a virtual separatrix presents the possibility of a computed trajectory actually jumping through it due to the finite time-steps of discrete numerical methods. Such behavior violates the uniqueness theory of differential equations and amplifies the numerical errors explosively. These reasons imply that, even if computed results are bounded, their independence on time-step should be established before accepting them as useful numerical approximations to the true solution of the differential equations. However, due to these exponential and explosive amplifications of numerical errors, no computed chaotic solutions of differential equations independent of integration-time step have been found. Thus, reports of computed non-periodic solutions of chaotic differential equations are simply consequences of unstably amplified truncation errors, and are not approximate solutions of the associated differential equations.

Lun-Shin Yao

Discrete numerical methods with finite time-steps represent a practical technique to solve initial-value problems involving nonlinear differential equations. These methods seem particularly useful to the study of chaos since no analytical chaotic solution is currently available. Using the well-known Lorenz equations as an example, it is demonstrated that numerically computed results and their associated statistical properties are time-step dependent. There are two reasons for this behavior. First, chaotic differential equations are unstable so that any small error is amplified exponentially near an unstable manifold. The more serious and lesser-known reason is that stable and unstable manifolds of singular points associated with differential equations can form virtual separatrices. The existence of a virtual separatrix presents the possibility of a computed trajectory actually jumping through it due to the finite time-steps of discrete numerical methods. Such behavior violates the uniqueness theory of differential equations and amplifies the numerical errors explosively. These reasons imply that, even if computed results are bounded, their independence on time-step should be established before accepting them as useful numerical approximations to the true solution of the differential equations. However, due to these exponential and explosive amplifications of numerical errors, no computed chaotic solutions of differential equations independent of integration-time step have been found. Thus, reports of computed non-periodic solutions of chaotic differential equations are simply consequences of unstably amplified truncation errors, and are not approximate solutions of the associated differential equations.

Lun-Shin Yao

There are many subtle issues associated with solving the Navier-Stokes equations. In this paper, several of these issues, which have been observed previously in research involving the Navier-Stokes equations, are studied within the framework of the one-dimensional Kuramoto-Sivashinsky (KS) equation, a model nonlinear partial-differential equation. This alternative approach is expected to more easily expose major points and hopefully identify open questions that are related to the Navier-Stokes equations. In particular, four interesting issues are discussed. The first is related to the difficulty in defining regions of linear stability and instability for a time-dependent governing parameter; this is equivalent to a time-dependent base flow for the Navier-Stokes equations. The next two issues are consequences of nonlinear interactions. These include the evolution of the solution by exciting its harmonics or sub-harmonics (Fourier components) simultaneously in the presence of a constant or a time-dependent governing parameter; and the sensitivity of the long-time solution to initial conditions. The final issue is concerned with the lack of convergent numerical chaotic solutions, an issue that has not been previously studied for the Navier-Stokes equations. The last two issues, consequences of nonlinear interactions, are not commonly known. Conclusions and questions uncovered by the numerical results are discussed. The reasons behind each issue are provided with the expectation that they will stimulate interest in further study.

Lun-Shin Yao

All complex fluid motions, such as transition and turbulence, obeying the Navier-Stokes equations are non-linear phenomena. Some aspects of the non-linear terms of these equations are not well understood and are, in fact, misunderstood. The one-dimensional Kuramoto-Sivashinsky (KS) equation is used as a simple model non-linear partial differential equation to show some essential functions of its non-linear term and its consequences, which, we believe, are shared with other non-linear partial differential equations. We show that solutions of nonlinear partial differential equations above their critical parameters may be linearly stable, but are nonlinearly unstable. No stable solution exists above the critical parameter, contrary to the prediction of the linear-stability analysis. This is because a linearly stable disturbance can transfer energy simultaneously, not necessarily in cascade from small wave numbers to large wave numbers. An initial disturbance can breed its entire harmonics simultaneously. Second, I show that a long-time numerical chaotic solution cannot be achieved by a discrete numerical method.