Abstract

In this talk we go back to the very basics of numerical analysis of PDEs, stability theory of finite difference schemes for linear evolution equations with variable coefficients. We prove that a universal "magic wand" renders numerical methods stable: the (first) space derivative should be discretised by a skew-symmetric matrix. The downside, however, is a barrier of 2 on the order of such methods on uniform grids. We derive a general theory coupling grid structure with the availability of skew-symmetric matrices corresponding to high-order methods.