# Goal Oriented Time Adaptivity Using Local Error Estimates
## 1 Introduction  
A typical situation in numerical simulations based on differential equations is that one is not interested in the solution per se but in a _Quantity of Interest_ (QoI) that is given as a functional of the solution. When designing an airplane, the QoI would be the lift coefficient divided by the drag coefficient. In simulations of the Greenland ice sheet, it could be the net amount of ice loss over a year [1]. For wind turbine design, the amount of energy produced during a certain time period is more important than the actual flow solution. Other examples are in optimization with ODEs or PDEs as constraints. In the turbine example, one may want to optimize blade shape [2] or determine optimal placement of, for example, tidal turbines [3], for maximal energy output. In this article, we consider time-dependent problems with a QoI as follows:  
\[\text{IVP}: \quad\dot{\mathbf{u}}(t)=\mathbf{f}(t,\mathbf{u}(t)),\quad t\in[t_{0},t_{e}],\quad\mathbf{u}(t_{0})=\mathbf{u}_{0},\quad\mathbf{f}:[t_{0},t_{e}]\times\mathbb{R}^{d} \rightarrow\mathbb{R}^{d} \tag{1}\] \[\text{QoI}: \quad\mathcal{J}(\mathbf{u}):=\int_{t_{0}}^{t_{e}}j(t,\mathbf{u}(t)) \text{d}t,\quad j:[t_{0},t_{e}]\times\mathbb{R}^{d}\rightarrow\mathbb{R} \tag{2}\]  
The goal is to find an adaptive time discretization to reduce computational costs. An _adaptive method_ consists of a _time-integration_ scheme for Equation (1), an _error estimator_, a _timestep controller_ and a discretization scheme for Equation (2). The input for an adaptive method is a tolerance \(\tau\) and initial timestep \(\Delta t_{0}\). The output is an approximation to the solution, this can be a discrete solution \(\mathbf{u}_{h}\approx\mathbf{u}\), or \(\mathcal{J}_{h}(\mathbf{u}_{h})\). **Definition 1**. : _An adaptive method for Equation (1) is called convergent, if_  
\[\|\mathbf{u}(t_{e})-\mathbf{u}_{N}\|\to 0\quad\text{for}\quad\tau\to 0,\]_where \(\mathbf{u}_{N}\approx\mathbf{u}(t_{e})\). An adaptive method for Equations (1) and (2) is called convergent in the QoI, if_  
\[|\mathcal{J}(\mathbf{u})-\mathcal{J}_{h}(\mathbf{u}_{h})|\to 0\quad\text{for}\quad \tau\to 0.\]  
The standard approach for goal oriented adaptivity is the dual weighted residual (DWR) method [4; 5]. It is most commonly used in the context of space-adaptivity for PDEs, but there is also work specifically on time-adaptivity [6; 7; 8]. DWR uses the adjoint (dual) problem to perform a posteriori adaptivity and get error bounds for the QoI. In the time dependent case, the adjoint problem is a terminal value problem, i.e., an IVP backwards in time. The method yields global error estimates (bounds for linear problems). These error estimates are obtained by subsequently integrating forward and backward in time, storing both solutions. The estimates contain local information used to refine the discretization. This iterative process is repeated until a discretization is found with an error estimate \(\eta(\mathbf{u}_{h})\) that fulfills  
\[|\mathcal{J}(\mathbf{u})-\mathcal{J}_{h}(\mathbf{u}_{h})|\approx\eta(\mathbf{u}_{h})\leq\tau.\]  
The resulting grids are typically of higher quality than those of local error based adaptive methods, particularly for advection dominated PDEs with spatially local QoIs. The major drawback of this method is its cost, both in implementation and computation. To reduce computational effort, [9] suggested to apply the approach in a block-wise manner, thus making it more local. The method requires a variational formulation and is thus typically used in the context of Galerkin type schemes but can also be used, for example, for linear multistep methods [10]. Alternatives are classical adaptive methods for IVPs (Equation (1)) based on local error estimation. Convergence results are well established and described in standard textbooks [11; 12]. They do not yield global error bounds. However, the error estimate is cheap and works very well for stiff problems, which typically have a dissipation mechanism to a base solution.