ADER scheme

Reservoir Flow Simulation by Adaptive ADER Schemes... 

In the following, we discuss in detail the construction of ADER schemes for linear and nonlinear problems. According to the WENO reconstruction procedure in Section 3, the solution u t, x) at the discrete time t = t is... 

It is well-known, that explicit time discretization schemes, such as the proposed ADER scheme, have to satisfy rather severe restrictions on the time step T due to the Courant-Friedrich-Levy (CFL) condition. Loosely speaking, disturbances from one cell boundary must not reach another boundary within one time step. Let p( be the radius of the inscribed circle of a triangular cell Ti serving as a measure of its diameter (see Figure 7), and let... 

In the last few years, ADER schemes were developed and analysed mainly for one-dimensional linear and nonlinear problems [36, 38, 39] and applied to multi-dimensional problems on fixed rectangular Cartesian meshes, e.g. in [30, 31]. Here we investigate the performance of the proposed ADER schemes for linear and nonlinear problems on unstructured triangulations by determining their convergence properties numerically. Furthermore, we consider their efEciency with respect to computational cost depending on the order of the scheme. 

In this section, the experimental order of convergence of the proposed ADER schemes on two-dimensional linear and nonlinear advection problems are determined numerically in order to compare them with the theoretically expected orders. 

In this subsection, we numerically evaluate the proposed ADER schemes with respect to computing time and achieved accuracy which should help to answer the above question. Therefore, we record the CPU time used by... 

Reservoir Flow Simulation by Adaptive ADER Schemes 365 Table 7. CPU seconds for the different computational step of ADER schemes. 

Table 8. Factors indicating the slowdown of ADER schemes.

Fig. 11. Accuracy of ADER schemes for different mesh width h, for ADERl (top), ADER2, ADER3, and ADERl (bottom).

In general, our results in Figure 12 show, that for problems with smooth solutions higher order schemes are more efficient than low order schemes with globally refined meshes, especially when highly accurate results are desired. In order to enhance the accuracy of the proposed ADER schemes even more, especially for solutions with discontinuities, we combine the ADER schemes with the ideas of adaptive mesh refinement in order to reduce numerical smearing. Details of our adaptive mesh strategy are discussed below. 

One important feature of our ADER schemes on unstructured triangulations is the time dependent adaptive mesh. Adaptivity requires the modification of the triangulation T during the simulation in order to be able to balance the two conflicting requirements of good approximation quality and small computational costs. In fact, for the sake of reducing the computational complexity we wish to reduce the number of cells, whereas for the sake of good approximation quality we prefer to use a fine mesh and therefore increase the number of cells. 

We have combined the proposed ADER schemes with the ideas of the adaption strategy, that has been discussed in previous work [5, 6, 18] and has proved to be efficient and robust. Here, we briefly review the basic ideas of the adaption strategy. 

Here, the scalar field 4> x) describes the porosity of the rock, the vector fields ayj t,x) and ao t,x) are the phase velocities, and Uw t,x) and Uo t,x) are the saturations of water and oil, respectively. Note, that Uy and Ug are the fractions of the pore space, that are filled with water or oil, i.e., 0 < Uw,o < 1-Equations (24) and (25) indicate, that a change of mass for each phase in a given region of a reservoir is equal to the net flux of the phase across the boundary of that region. Therefore, the class of finite volume schemes, such as the proposed ADER schemes, are obviously a natural choice from available numerical methods to solve such problems. 

In order to see the differences between low and high order ADER schemes, we first show the color-coded water saturation obtained with an adaptive ADERl scheme in Figure 16. The shocks representing the interface between pure oil and a mixture of oil and water are moving from the corners of the model reservoir towards its center. This way, oil in the porous medium is displaced by water, i.e., it is effectively pushed towards the production well. Before the shocks actually reach the production well at the center, the sucking effect of the production well becomes obvious, which is due to the increasing total velocity resulting from the pressure drop at the production well. 

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