SBP-FDEC: Summation-by-Parts Finite Difference Exterior Calculus
This work provides a new methodology for constructing compatible discretizations in computational physics, particularly for problems requiring exact preservation of divergence and curl constraints.
The authors extend the Finite Element Exterior Calculus (FEEC) framework to Summation-by-Parts Finite Difference (SBP-FD) methods, enabling divergence- and curl-free discretizations. They achieve this by leveraging a novel analytic relationship between integral and nodal degrees of freedom, using pre-existing SBP-FD operators.
We demonstrate that we can carry over the strategy of Finite Element Exterior Calculus (FEEC) to Summation-by-Parts (SBP) Finite Difference (FD) methods to achieve divergence- and curl-free discretizations. This is not obvious at first sight, as for SBP-FD no basis functions are known, but only values and derivatives at points. The key is a remarkable analytic relationship that enables us to construct compatible operators using integral and nodal degrees of freedom. Pre-existing SBP-FD matrix operators can then be used to obtain nodal values from the integral degrees of freedom to derive a scheme with the desired properties.