Finite Element Hodge for Spline Discrete Differential Forms. Application to the Vlasov-Poisson Equations

The notion of B-spline based discrete differential forms is recalled and along with a Finite Element Hodge operator, it is used to design new numerical methods for solving the Vlasov-Poisson equations. Answer’s to the referees’ questions Aurore Back and Eric Sonnendrücker April 12, 2013 We thank the referees for the careful reading of our manuscript and for many interesting comments that helped us improve the article. 1 Answer to the questions of referee 0 1. We agree that exterior calculus is not really necessary in 1D, but this work provides a proof of principle that discrete exterior calculus tools can be applied to the simulation of Vlasov-Poisson and Vlasov-Maxwell systems. Discrete exterior calculus has been proven extremely beneficial for the simulation of the Maxwell equations and this will of course extend to the Vlasov-Maxwell equations, bringing all the benefits for Maxwell equations and in addition some very useful properties in particular for the coupling of Vlasov with Maxwell with the problem of discrete charge conservation is an important and challenging issue. This discussion has been added in the introduction. 2. The tensor product extension to higher dimensions of the Hodge operator is straightforward and present no difficulty. 3. The equivalence of Ampère and Poisson is true only in 1D. A justification is provided in the appendix. 4. Done. 5. Done. 6. Yes, section 4.1 is devoted to expressing the most important conservation properties at the continuous level in the language of differential forms. This is conservation of mass, momentum and energy. These are not automatically conserved at the discrete level even using discrete differential forms, but it can be obtained by choosing carefully the discretisation spaces and the Poisson solver with a full 2D scheme. For cost reasons, we use here a 1D splitting in which case there are no discrete conservations of momentum and energy. The discrete obtained in our framework is centred and not diffusive. This is explained. 1 Response to Reviewers