Stability

An algorithm for solving an evolutionary partial differential equation is said to be stable if the numerical solution at a fixed time remains bounded as the step size goes to zero, so the pertrubations in form of, e.g., rounding error does not increase in time. Unfortunately, there are no general methods to verify the numerical stability for the partial differential equations in general form, so one restrict oneself to the case of linear PDE's. The standard method for linear PDE's was proposed by John von Neumann in 1947 and is based on the representation of the rounding error in form of the Foirer series.