Summation by Parts Operators for Lagrangian Field Evaluations

In the aim of computing the evolution of electromagnetic potentials for theoretical work, it is common practice to produce wave equations from the analytic Lagrangian and discretize the vector calculus operators post hoc. For most choices of finite difference operators, this results in a set of discretized equations which do not correspond to the discretized version of the Lagrangian. By choosing discrete analogs to derivative and integral operations which preserve integration by parts, so-called Summation by Parts operators [1], the discretized electromagnetic potentials may be derived directly from a discretized Lagrangian and can be rigorously analyzed using the wealth of techniques common to Lagrangian mechanics. We illustrate these considerations with a model of a charged beam traveling through a plasma.

[1] H Ranocha, K. Ostaszewski, and P. Heinish. “Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators.” Communication on Applied Mathematics and Computation, 2020. (https://doi.org/10.1007/s42967-019-00057-2)