# Goal Oriented Time Adaptivity Using Local Error Estimates
## 5 Numerical Results
### Thermal Fluid Structure Interaction  
The previous examples were of academic nature, to demonstrate strength and weaknesses of the method, as predicted in Section 4.3. As a third test problem, we consider the coupling of two heat equations with different thermal conductivities and diffusivities. This models, for example, steel  
Figure 4: Comparison of the various methods for the convection diffusion Problem (34) with QoI as in Equation (35). (**Top**): Performance for positive \(\mathbf{v}\). (**Centre**): Tolerance over error and chosen timesteps for \(\tau=10^{-6}\) (positive \(\mathbf{v}\)). (**Bottom**): Performance for negative \(\mathbf{v}\) and \(t_{e}=3\).  
cooling with pressurized air. Furthermore, this example will show how the proposed method can be applied seamlessly in a partitioned approach. The model equations for this problem are  
\[\alpha_{m}\frac{\partial u_{m}(t,\mathbf{x})}{\partial t}-\
abla\cdot( \lambda_{m}\
abla u_{m}(t,\mathbf{x}))=0, (t,\mathbf{x})\in[t_{0},t_{e}]\times\Omega_{m},\] \[u(t,\mathbf{x})=0, (t,\mathbf{x})\in[t_{0},t_{e}]\times\Omega_{m}\setminus\Gamma,\] \[u_{1}(t,\mathbf{x})=u_{2}(t,\mathbf{x}), \lambda_{2}\frac{\partial u_{2}(t,\mathbf{x})}{\partial n_{2}}=- \lambda_{1}\frac{\partial u_{1}(t,\mathbf{x})}{\partial n_{1}}, (t,\mathbf{x})\in[t_{0},t_{e}]\times\Gamma, \tag{36}\] \[u_{m}(t_{0},\mathbf{x})=u_{m}^{0}(\mathbf{x}), \mathbf{x}\in\Omega_{m},\quad m=1,2.\]  
Our QoI is the time-averaged heat transfer over the interface \(\Gamma\) and given by  
\[\mathcal{J}(u)=\int_{t_{0}}^{t_{e}}\int_{\Gamma}\frac{1}{t_{e}-t_{0}}\lambda_ {1}\frac{\partial u_{1}(t,\mathbf{x})}{\partial n_{1}}\ \mathrm{d}\mathbf{x}\,\mathrm{d}t,\]  
which in the discrete case is given by the finite difference  
\[\mathcal{J}_{h}(u_{h})=\sum_{i=1}^{n-1}u_{\Gamma}^{i}-u_{1,\Gamma}^{i},\]  
where \(u_{\Gamma}^{i}\approx u(1,i\,\Delta x)\) and \(u_{1,\Gamma}^{i}\approx u(1-\Delta x,i\,\Delta x)\).  
We consider the spatial domains \(\Omega_{1}=[0,1]\times[0,1]\), \(\Omega_{2}=[1,2]\times[0,1]\), \(\Gamma=\Omega_{1}\cap\Omega_{2}\), and \([t_{0},t_{e}]=[0,0.2]\). We use the initial conditions  
\[u^{0}(x,y)=\begin{cases}800\sin(\pi y)^{2}\sin(2\pi x),&x\leq 0.5\\ 200\sin(\pi y)^{2}\sin(\pi(x-0.5)),&0.5<x\leq 1.5\\ 0,&\text{else}\end{cases}, \tag{37}\]  
see Figure 5, which also shows the geometry.  
For time-integration we use SDIRK2, which is implicit with \((p,\,\hat{p})=(2,\,1)\). To solve the problem arising from the so called _transmission conditions_ (Equation (36)) on the interface \(\Gamma\), we use the Dirichlet-Neumann iteration for each stage derivative of SDIRK2 [29]. We choose the parameters \(\alpha_{1}=0.1,\ \lambda_{1}=0.01\) and \(\alpha_{2}=\lambda_{2}=1\). Based on the results of [30], this gives us a convergence rate of approximately \(\alpha_{1}/\alpha_{2}=0.1\) for the Dirichlet-Neumann iteration for \(\Delta t\to 0\).  
Standard linear FE are used in space for both domains with identical triangular meshes for \(\Delta x=1/81\). Details of the implementation of the discretization and methods can be found in [31].  
For this problem, we use \(\Delta t_{0}=\tau\). The reference solution is computed using the classical scheme with \(\tau=10^{-6}\) and an absolute tolerance of \(10^{-12}\) in the Dirichlet-Neumann iteration, where the numerical tests use an absolute tolerance of \(\tau/100\).  
Figure 5: (**Left**): Geometry for the coupled heat problem. (**Right**): Initial condition (Equation (37)).