A Partial Nonlinear Extension of Lax-Richtmyer Approximation Theory

Lax and Richtmyer developed a theory of algorithms for linear initial value problems that guarantees, under certain circumstances, the convergence to numerical solution of initial value problem. The assumptions are first that the difference equations (algorithms) approx- imate the differential equations under study (this is called consistency) and, secondly, that the initial value problem be well-posed (which means that the solutions exist, are unique and depend continuously on initial data). Under these assumptions the stability condition (which requires that errors in the algorithm do not accumulate nor increase as one iter- ates the algorithm) is necessary and sufficient for convergence in a certain uniform sense for arbitrary initial data. In this work we will extend certain aspects of their work to the nonlinear context. We drop the PDE and the well-posedness assumptions at first and add the ”β − axioms” that will guarantee convergence [ Theorems 2 and 3 ] of algorithm orbits in a projective limit of finite dimensional spaces. A conjecture for a partial converse that some stability is a consequence of convergence for a natural class of nonlinear algorithms where the deviation of these non-linear algorithms from being linear is itself a bilinear map. When the algorithms satisfy consistency with a PDE initial value problem we obtain the definition of a new kind of numerical solution and their existence [Theorem 6] given said algorithms