Spatially mass-, kinetic energy- and helicity-preserving mimetic discretization of 3D incompressible Euler flows

Abstract

Based on the recently developed mimetic spectral element method, we propose an effective numerical scheme for solving three-dimensional periodic incompressible Euler flows, which spatially preserves mass, kinetic energy and helicity. Preserving multiple integral invariants numerically will significantly contribute to the stability and accuracy of the numerical scheme. We start from the introduction of differential geometry and algebraic topology with which we then set up the mimetic spectral element method (the mimetic framework). With the mimetic spectral element method, physical variables can be expressed in more physical forms and the discretization error will be eliminated as much as possible. After that, we turn to Euler equations. We first rewrite Euler equations as inner oriented Euler equations and outer oriented Euler equations in terms of differential differential forms. Meanwhile the conservation laws of mass, kinetic energy and helicity based on these new forms of Euler equations at the continuous level are proven. Then, according to expressions of kinetic energy and helicity in the mimetic framework, we convert the inner oriented and outer oriented Euler equations into two weak forms and then spatially discretize them using the mimetic spectral element method in a unit 3-cube ([-1,1]^{3}) domain equipped with a cell complex given by the Gauss-Lobatto-Legendre grid. Afterwards, interactions between the two spatially discretized weak forms and discretizations of time derivative terms are constructed, which eventually gives rise to a solvable, mass, kinetic energy and helicity spatially preserved, fully discretized system. The scheme then is tested with a periodic flow. In summary, the mass conservation is automatically achieved by taking the divergence free flow condition exactly into account. In addition, with proper discretization of the momentum equation and vorticity equation both kinetic energy and helicity are spatially preserved at the discretization level.