论文标题
非线性项在不可压缩流的数值近似中的影响通过正确的正交分解方法
On the influence of the nonlinear term in the numerical approximation of Incompressible Flows by means of proper orthogonal decomposition methods
论文作者
论文摘要
我们考虑适当的正交分解（POD）方法来近似不可压缩的Navier-Stokes方程。我们研究了在快照中使用非线性项的一个离散化（使用完整阶方法（FOM）计算的），并且在POD方法中应用非线性项的不同离散化。与在这种情况下，我们证明了一个额外的错误项，与在FOM和POD方法中同时应用非线性项的离散化相比。但是，附加项的大小与FOM的误差相同，因此POD方法的收敛速率几乎没有影响。我们分析了我们在FOM和POD方法中添加Grad-Div稳定的情况，因为它允许与粘度的逆强度获得常数的误差界限。我们还研究了未添加稳定的情况。一些数值实验支持理论分析。
We consider proper orthogonal decomposition (POD) methods to approximate the incompressible Navier-Stokes equations. We study the case in which one discretization for the nonlinear term is used in the snapshots (that are computed with a full order method (FOM)) and a different discretization of the nonlinear term is applied in the POD method. We prove that an additional error term appears in this case, compared with the case in which the same discretization of the nonlinear term is applied for both the FOM and the POD methods. However, the added term has the same size as the error coming from the FOM so that the rate of convergence of the POD method is barely affected. We analyze the case in which we add grad-div stabilization to both the FOM and the POD methods because it allows to get error bounds with constants independent of inverse powers of the viscosity. We also study the case in which no stabilization is added. Some numerical experiments support the theoretical analysis.
