The Geometry of Time-stepping

The space-time geometric structure of Maxwell’s equations is examined and a subset of them is found to define a pair of exact discrete time-stepping relations. The desirability of adopting an approach to the discretization of electromagnetic problems which exploits this fact is advocated, and the name topological time-stepping for numerical schemes complying with it is suggested. The analysis of the equations leading to this kind of time-stepping reveals that these equations are naturally written in terms of integrated field quantities associated with space-time domains. It is therefore suggested that these quantities be adopted as state variables within numerical methods. A list of supplementary prescriptions for a discretization of electromagnetic problems suiting this philosophy is given, with particular emphasis on the necessity to adopt a space-time approach in each discretization step. It is shown that some existing methods already comply with these tenets, but that this fact is not explicitly recognized and exploited. The role of the constitutive equations in this discretization philosophy is briefly analyzed. The extension of this approach to more general kinds of space-time meshes, to other sets of basic time-stepping equations and to other field theories is finally considered.