A Numerical Predictability Problem in Solution of the Nonlinear Diffusion Equation

A numerical analysis of the nonlinear heat diffusion equation has been performed to bring to light a type of instability that can be encountered in many numerical modelling 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 upon the Richardson number, which, in turn, is a function of the dependent variable. The analysis shows that the interaction of short-wavelength and intermediate-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 by using a boundary layer model with GATE Phase 3 data are presented to support the theoretical conclusions.