A nonlinear multigrid steady-state solver for 1D microflow

Abstract

We develop a nonlinear multigrid method to solve the steady state of microflow, which is modeled by the high order moment system derived recently for the steadystate Boltzmann equation with ES-BGK collision term. The solver adopts a symmetric Gauss-Seidel iterative scheme nested by a local Newton iteration on grid cell level as its smoother. Numerical examples show that the solver is insensitive to the parameters in the implementation thus is quite robust. It is demonstrated that expected efficiency improvement is achieved by the proposed method in comparison with the direct timestepping scheme.

An efficient steady-state solver for microflows with high-order moment model

“…The collision frequency for the variable hard sphere model, that is, with the viscosity index w = 0.5 and the Knudsen number Kn = 0.1 is adopted. With these settings, the steady-state solutions for density ρ, temperature θ, normal stress σ 11 and heat flux q 2 , obtained by the NMLM solver on the uniform grid with N = 200, are shown in Figure 9, which coincide well with the steady-state solutions presented in [21], where the first-order spatial discretization with N = 2048 is employed.…”

“…Based on our numeircal experience, there are several advantages by using Heun's method. Comparing to the SGS-Newton iteration proposed in [21], Heun's method can be implemented much easier, while comparing to the SGS-Richardson iteration proposed in [22], Heun's method exhibits better performance, especially when a large Knudsen number is considered. It is worth mentioning that Heun's method would enhance the robustness of the NMLM solver.…”

Numerical Simulation of Microflows Using Hermite Spectral Methods

“…The SGS iteration is in general several times faster than the explicit time-stepping scheme, although for both methods, the total number of iterations is expected to grow linearly as the grid number increases. Further acceleration of the steady-state computation can be obtained by using the multigrid technique, which has been explored in [19,21]. The same framework of the nonlinear multigrid method as proposed in [19] is used in our simulation, except that the single level iteration is replaced by the above SGS iteration.…”

“…Further acceleration of the steady-state computation can be obtained by using the multigrid technique, which has been explored in [19,21]. The same framework of the nonlinear multigrid method as proposed in [19] is used in our simulation, except that the single level iteration is replaced by the above SGS iteration. By noting that u and θ are constants, 4 the implementation is in fact much easier than that in [19].…”

Acceleration for microflow simulations of high-order moment models by using lower-order model correction

“…To accomplish the multi-level moment algorithm, the framework of nonlinear multigrid algorithm developed in [15] would be used. The implementation follows the basic idea of the NRxx method, such that the resulting nonlinear multi-level moment (NMLM) solver also has a uniform framework for the model of arbitrary order, and has the same input and output interfaces as the NMG solver introduced in [19]. Moreover, the transformation operators between models of different orders could be implemented efficiently under the framework of the NRxx method.…”