In this paper, we analyze standard WENO reconstructions and multilevel WENO reconstructions with adaptive order (WENO-AO) using both WENO-JS and WENO-Z weighting. We also present a new WENO-AO reconstruction. We give conditions under which the reconstructions achieve optimal order accuracy for both smooth solutions and solutions with discontinuities. The old WENO-AO reconstruction drops to a fixed, base level of approximation when there are discontinuities in the solution, but the new one maintains the accuracy of the largest stencil over which the solution is smooth. Our analysis in the discontinuous case requires that the smoothness indicators do not approach zero as the grid is refined. We provide a condition to ensure this result, but we also show an example where this can fail to occur. That is, we show that WENO reconstructions can fail to maintain the order of approximation of the smallest stencil over which the solution is smooth. We also present numerical results confirming the convergence theory of the old and new WENO-AO reconstructions, and compare their performance in solving conservation laws.ARBOGAST, HUANG, AND ZHAO polynomials of different degrees, e.g., as done by Cravero, Puppo, Semplice, and Visconti [5]. The idea was further generalized by Zhu and Qiu [15] and Balsara, Garain, and Shu [2] to define the class of multilevel WENO reconstructions with adaptive order (WENO-AO).In this paper, we analyze the accuracy of the standard WENO and WENO-AO reconstructions. We include results for when the WENO-Z weighting procedure of Castro, Costa, and Don [4, 3] is used. Our results are summarized in Sections 4.3.5 and 5.3 below. The standard WENO reconstruction behaves as desired, even with WENO-Z weights. It is high order accurate when the solution is smooth, and it drops to low order s when there is a discontinuity (not on the central cell), provided only that η ≥ s/2. We show that this condition is sharp. The two-level WENO-AO(r, s) reconstructions behave similarly. They can achieve higher order r accuracy in the smooth case and otherwise maintain at least order s accuracy, provided that when using WENO-JS weights, r ≤ 2s − 1 and η ≥ s/2. WENO-Z weights are more complex, because the accuracy of the reconstruction depends more strongly on the choice of η, as we will show.Multilevel WENO-AO s (r , . . . , r 1 , s) reconstructions [2] are based on a base state with approximation order s. When the solution is smooth the approximation attains the highest order r , but when it is discontinuous, it usually drops to the base level s. When using WENO-JS weights, one requires r ≤ 2s − 1 and η ≥ s/2, which is equivalent to the two-level case. That is, from the point of view of approximation order, there is no point in using the multilevel reconstruction. Again, WENO-Z weights are more complex. There is a restriction on the size of the gap between successive approximation levels, but a careful choice of η can allow any order. However, the accuracy almost always drops to order s when the solution is discontinuous....

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Simulation of flow and transport in petroleum reservoirs involves solving coupled systems of advection diffusion-reaction equations with nonlinear flux functions, diffusion coefficients, and reactions/wells. It is important to develop numerical schemes that can approximate all three processes at once, and to high order, so that the physics can be well resolved. In this paper, we propose an approach based on high order, finite volume, implicit, Weighted Essentially NonOscillatory (iWENO) schemes. The resulting schemes are locally mass conservative and, being implicit, suited to systems of advection-diffusion reaction equations. Moreover, our approach gives unconditionally L-stable schemes for smooth solutions to the linear advection-diffusion-reaction equation in the sense of a von Neumann stability analysis. To illustrate our approach, we develop a third order iWENO scheme for the saturation equation of two-phase flow in porous media in two space dimensions. The keys to high order accuracy are to use WENO reconstruction in space (which handles shocks and steep fronts) combined with a two-stage Radau-IIA Runge-Kutta time integrator. The saturation is approximated by its averages over the mesh elements at the current time level and at two future time levels; therefore, the scheme uses two unknowns per grid block per variable, independent of the spatial dimension. This makes the scheme fairly computationally efficient, both because reconstructions make use of local information that can fit in cache memory, and because the global system has about as small a number of degrees of freedom as possible. The scheme is relatively simple to implement, high order accurate, maintains local mass conservation, applies to general computational meshes, and appears to be robust. Preliminary computational tests show the potential of the scheme to handle advection-diffusion-reaction processes on meshes of quadrilateral gridblocks, and to do so to high order accuracy using relatively long time steps. The new scheme can be viewed as a generalization of standard cell-centered finite volume (or finite difference) methods. It achieves high order in both space and time, and it incorporates WENO slope limiting.

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