Parametrization of the cumulant lattice Boltzmann method for fourth order accurate diffusion part I: Derivation and validation

The cumulant lattice Boltzmann method offers a set off free relaxation parameters that do not influence the result at leading order but can be used to influence the leading error. Using Taylor expansion we derive exact functional relationships for the elimination of the linearized leading error of the method. The diffusion term in the Navier-Stokes equation becomes fourth order accurate for small enough viscosity with these parameters. The result is general and does not depend on the flow. The analytical solution is tested against Taylor-Green vortex flow and shear wave flow and fourth order accuracy for the diffusion is observed.