Title: Removing the numerical instability of PDE’s
Abstract: I present a general method to remove the numerical instability of partial differential equations. It is based on the idea that only a rough approximation to an equation’s behavior at small scales is necessary to remove instability, while retaining the same level of accuracy a fully implicit treatment would have. Thus we can choose stabilizing terms which can be inverted efficiently, allowing the treatment of problems which would otherwise be prohibitively expensive. Based on the same insight, I also present a method which chooses stabilizing terms adaptively, so as to supress numerical noise on small scales. This removes the necessicity of analyzing the small-wavelenght behavior, and chooses stabilizing terms optimally.