A Numerical Predictability Problem in Solution of the Nonlinear Diffusion Equation

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Abstract

A numerical analysis of the nonlinear heat diffusion equation has been carted out to bring to light a heretofore little-understood type of instability that can be encountered in many numerical modeling applications. The nature of the instability is such that the error remains bounded but becomes large enough to prevent proper assessment of model results. For the sample problem under investigation, the nonlinearity is introduced through a diffusion coefficient that depends on the Richardson number which, in turn, is a function of the dependent variable. Our analysis shows that the interaction of short-wavelength and inter-mediate-wavelength solution components can induce nonlinear instability if the amplitude of either component is sufficiently large. Since the unstable solution may not wander far from the true solution, the error can be difficult to detect. A criterion, given in terms of a restriction on the Richardson number, guarantees local (short-term) stability of the numerical scheme whenever the criterion is satisfied. Numerical results obtained using a boundary-layer model with GATE Phase III data are presented to support the theoretical conclusions.

Abstract

A numerical analysis of the nonlinear heat diffusion equation has been carted out to bring to light a heretofore little-understood type of instability that can be encountered in many numerical modeling applications. The nature of the instability is such that the error remains bounded but becomes large enough to prevent proper assessment of model results. For the sample problem under investigation, the nonlinearity is introduced through a diffusion coefficient that depends on the Richardson number which, in turn, is a function of the dependent variable. Our analysis shows that the interaction of short-wavelength and inter-mediate-wavelength solution components can induce nonlinear instability if the amplitude of either component is sufficiently large. Since the unstable solution may not wander far from the true solution, the error can be difficult to detect. A criterion, given in terms of a restriction on the Richardson number, guarantees local (short-term) stability of the numerical scheme whenever the criterion is satisfied. Numerical results obtained using a boundary-layer model with GATE Phase III data are presented to support the theoretical conclusions.

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