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A Monotone Petrov-Galerkin Method for Quasilinear Parabolic Differential Equations

To stabilize convection-dominated flows, artificial diffusion is often introduced by upstream weighting. A fundamental question is therefore: What amount of artificial diffusion should be introduced to avoid numerical oscillations? Obviously, that amount needs to be minimized, since artificial diffusion causes smearing of sharp fronts and less spatial accuracy. This paper analyzes the Petrov–Galerkin method with linear basis functions and quadratic, asymmetric test functions. The author uses the ideas of monotonicity to control the artificial diffusion introduced so that the numerical schemes are a priori monotone, i.e., numerical oscillations will not form if the discrete equations are solved exactly.