Artificial compressibility method revisited: Asymptotic numerical method for incompressible Navier-Stokes equations

The artificial compressibility method for the incompressible Navier-Stokes equations is revived as a high order accurate numerical method (4th order in space and 2nd order in time). Similar to the lattice Boltzmann method, the mesh spacing is linked to the Mach number. The accuracy higher than that of the lattice Boltzmann method is achieved by exploiting the asymptotic behavior of solution of artificial compressibility equations for small Mach numbers and the simple lattice structure of stencil. An easy method for quickening the decay of acoustic waves, which deteriorate the quality of numerical solution, and a simple cure for the checkerboard instability are proposed there. The high performance of the scheme is demonstrated not only for the periodic boundary condition but also for the Dirichlet-type boundary condition.