瞬态Navier-Stokes方程的一种新的全离散粘性稳定化方法

A New Full Discrete Stabilized Viscosity Method for the Transient Navier-Stokes Equations

基于压力投影和梯形外推公式,对速度/压力空间采用等阶多项式逼近,针对高Reynolds数下的瞬态Navier-Stokes方程提出了一种新的全离散粘性稳定化方法.该方法不仅绕开了inf-sup条件的限制,克服了高Reynolds数下对流占优造成的不稳定性,而且在每一时间步上,只需要进行线性计算,从而减少了计算量.给出了稳定性证明,并得出了与粘性系数一致的误差估计.理论和数值结果表明该方法具有二阶精度.

A new full discrete stabilized viscosity method for the transient Navier-Stokes equations with the high Reynolds number （small viscosity coefficient ） was proposed based on pressure projection and extrapolated trapezoidal rule. The transient Navier-Stekes equations are fully-discretized by continuous equal-order finite elements in space and reduced Crank-Nicolson scheme in time. The new stabilized method is stable and has a number of attractive properties. Firstly, the system is stable for the equal-order combination of discrete continuous velocity and pressure spaces because of adding a pressure projection tenn. Secondly, the artifical viscosity parameter was added to the viscosity coefficient as a stability factor, so the system is antidiffusion. Finally, the method requires only the solution of one linear system per time step. Stability and convergence of the method was proved. The error estimation results show that the method has second order accuracy, and the constant in the estimation is independent of the viscosity coefficient. The numerical results were given, which demonstmte the advantage of the method presented.