Time-Implicit High-Order Accurate Positivity-Preserving Discretizations for the Navier-Stokes and Navier-Stokes-Korteweg Equations

We extend the framework of the Karush-Kuhn-Tucker (KKT) limiter from second-order nonlinear scalar equations to complex systems of equations and construct time-implicit high-order accurate positivity-preserving local discontinuous Galerkin methods for the Navier-Stokes(NS) and Navier-Stokes-Korteweg(NSK) equations, which are second-order and third-order nonlinear systems, respectively. For the NS and NSK equations, a meaningful numerical approximation must ensure that the density and pressure are non-negative. Borrowing from the idea of the KKT limiter, we use Lagrange multipliers to couple the positivity constraints of density and pressure with local discontinuous Galerkin discretizations combined with a diagonally implicit Runge-Kutta time integration method and obtain high-order accurate positivity-preserving schemes for the NS and NSK equations. The use of diagonally implicit Runge-Kutta time integration methods greatly enlarges the time step in the numerical simulation of the NS and NSK equations. Different from the original application of the KKT limiter, the nonlinearity of pressure with respect to the conservative variables increases the difficulty in implementation. Numerical examples validate the efficiency of these methods.