Energy and helicity preserving schemes for hydro- and magnetohydro-dynamics flows with symmetry

We propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry. By introducing a generalized vorticity-stream formulation, the divergence free constraints are automatically satisfied. In addition, with explicit treatment of the nonlinear terms and local vorticity boundary condition, the Navier–Stokes (MHD, respectively) equation essentially decouples into 2 (4, respectively) scalar equation and thus the scheme is very efficient. Moreover, with proper discretization of the nonlinear terms, the scheme preserves both energy and helicity identities numerically. This is achieved by recasting the nonlinear terms (convection, vorticity stretching, geometric source, Lorentz force and electro-motive force) in terms of Jacobians. This conservative property is valid even in the presence of the pole singularity for axisymmetric flows. The exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity. Numerical examples have demonstrated both accuracy and efficiency of the scheme. Finally, local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost.