  The flux reconstruction (FR) method has gained popularity in the research
community as it recovers promising high-order methods through modally filtered
correction fields, such as the discontinuous Galerkin method, amongst others,
on unstructured grids over complex geometries. Moreover, FR schemes,
specifically energy stable FR (ESFR) schemes also known as
Vincent-Castonguay-Jameson-Huynh schemes, have proven attractive as they allow
for design flexibility as well as stability proofs for the linear advection
problem on affine elements. Additionally, split forms have recently seen a
resurgence in research activity due to their resultant nonlinear (entropy)
stability proofs. This paper derives for the first time nonlinearly stable ESFR
schemes in split form that enable nonlinear stability proofs for, uncollocated,
modal, ESFR split forms with different volume and surface cubature nodes. The
critical enabling technology is applying the splitting to the discrete
stiffness operator. This naturally leads to appropriate surface and numerical
fluxes, enabling both entropy stability and conservation proofs. When these
schemes are recast in strong form, they differ from schemes found in the ESFR
literature as the ESFR correction functions are incorporated on the volume
integral. Furthermore, numerical experiments are conducted verifying that the
new class of proposed ESFR split forms is nonlinearly stable in contrast to the
standard split form ESFR approach. Lastly, the new ESFR split form is shown to
obtain the correct orders of accuracy.
