# Diverged HDG solution for 2D incompressible Navier-Stokes test case at SMALL time step. Why?

I wrote a hybridizable Discontinuous Galerkin code for transient Navier-Stokes flow, with thanks to Martin Kronbichler and Scott Miller's step-51 code in Deal.II. The main algorithm is from Nguyen and Peraire's JCP paper(2012).

$$\frac{1}{\nu}q_{ij}+\frac{\partial u_j}{\partial x_i}=0\;\;in \;\Omega$$ $$\frac{\partial u_j}{\partial t}+\frac{\partial}{\partial x_i}(q_{ij}+p\delta_{ij}+u_i u_j)=f_{j}\;\;in \;\Omega$$ $$\frac{\partial u_j}{\partial x_j}=0\;\;in \;\Omega$$ $$n_i(q_{ij}+p\delta_{ij})|_{\Gamma_N}=g^N_j$$ $$u_j|_{\Gamma_D}=g^D_j$$ $$u_j|_{t=0}=u^0_j$$ The weak form I've taken is as follows, $$(e_{ij},\frac{1}{\nu}q_{ij})_{T_h}-(\frac{\partial e_{ij}}{\partial x_i},u_j)_{T_h}+<n_ie_{ij},\hat{u}_j>_{\partial T_h}=0$$ $$(v_j,\frac{\partial u_j}{\partial t})_{T_h}+(v_j,\frac{\partial}{\partial x_i}(q_{ij}+p\delta_{ij}))_{T_h}-(\frac{\partial v_j}{\partial x_i},u_i u_j)_{T_h}+<v_j,n_i\hat{u}_i \hat{u}_j>_{\partial T_h}+<v_j,S(u_j-\hat{u}_j)>_{\partial T_h}=(v_j,f_j)_{T_h}$$ $$-(\frac{\partial r}{\partial x_j},u_j)_{T_h}+<r,n_j \hat{u}_j>_{\partial T_h}=0$$ $$(1,p)_{K}=\bar{p}A_K,\;\;\forall K\in T_h$$ and globally, $$<\mu_j,n_iq_{ij}+pn_j+S(u_j-\hat{u}_j)>_{\partial T_h}=<\mu_j,g^N_j>_{\Gamma_N}$$ $$<1,\hat{u}_jn_j>_{\partial T_h}=0$$ $$(1,\bar{p})_{T_h}=0\;\;(if\; \Gamma_N=\emptyset)$$ Now my code was tested with the 2D lid-driven cavity case, up to Re=1000. All tests passed with steady mode. But once the time derivatives are turned on, the code only converged to the steady result with large time step,e.g. dt=100. If the time step is small, such as 1.e-2, the solution diverged. In all cases, the stablization coefficient tau is kept constant as 5.0 .

Anyway, the less the time step, the more unstable the code. This really contradicts againt our common understanding of instability and convergence.

For simplicity, the time marching was done by backward Euler scheme. The Picard iteration was used for linearization.

I am just wondering if there's any other aspect in the stability of HDG method I should pay attention to.

Any possible suggestion will be appreciated.

• If you think there is something with the method, my suggestion would be probably to try it for a linear case to exlude the nonlinearity affecting the stability. – VorKir Dec 9 '16 at 20:14
• I found several paper on transient Stokes problems. There they mentioned about the so-called inverse CFL condition, which needs $$h^2\leq\nu\tau$$ May this cause the above problem? Anyone can talk more about it? Cheers. – Zhenyu Zhang Dec 10 '16 at 13:48
• Out of curiosity, could you give a reference? (completely not a specialist in Stokes) – VorKir Dec 11 '16 at 1:53
• see Burman & Fernández, Analysis of the PSPG method for the transient Stokes’ problem, 《Computer Methods in Applied Mechanics & Eng. , 2011, 200(41):2882-2890 – Zhenyu Zhang Dec 12 '16 at 6:38