The combination of suitable CFL conditions and the use of the high-order limiter proposed in [49] is sufficient to ensure positivity of the distribution function.
The design of the high-order limiter is guided by the findings of Zhang and Shu (2011) [36], and is based on limiting the quadrature values of the density, pressure and order parameters reconstructed using a high-order WENO scheme.
The formulation is conservative, and monotonicity is enforced by high-order limiters and by appropriately lowering the basis order around discontinuities.
The scheme is based on high-order weighted-essentially non-oscillatory (WENO) finite volume schemes and high-order limiters to ensure the maximum principle or positivity of the various field variables including the density, pressure, and order parameters identifying each phase.
It is quite possible that reconstruction is done with a second order TVD limiter, like the MC limiter.
For the spatial reconstruction, we adopt the second order TVD limiter due to Koren (Koren 1993).
The desired positivity-preserving (PP) property of the scheme is further attained by incorporating a newly proposed high order PP limiter.
In this work we propose a second order flux limiter finite volume method, named PVM-2U-FL, that only uses information of the two external waves of the hyperbolic system.
For the latter case, we combine the method so derived with Runge Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients.
The numerical solution approach comprises the discrete velocity method in the velocity space and the finite volume method in the physical space with different numerical flux schemes: the first-order, the second-order minmod flux limiter and a third-order WENO schemes.
In this concept, a number of torque limiters are included in the structure of the robot in order to limit the feasible forces at the end-effector.
