A TVD principle and conservative TVD schemes for adaptive Cartesian grids

Modern high-resolution conservative numerical schemes (see [1–3]) are widely used for simulating flows of liquids, gases, or plasmas. They provide a robust and accurate method for capturing discontinuities, such as shock waves. A total-variation-diminishing (TVD) principle is an important element of such schemes. A TVD scheme is constructed in such a way that the total variation, TV 1⁄4 P ikUi Ui 1k, would not increase as the numerical solution is advanced in time. The TVD property of the scheme does not allow the appearance of new local extrema for the grid function Ui, eliminating any spurious oscillations, that tend to form, e.g., behind a shock wave. For uniform Cartesian grids, the TVD property of the high-order scheme can be achieved by employing a Riemann solver together with a limiting procedure for the gradients of the variables. The limiting procedure yields the reconstructed face values that are inputs for the Riemann solver. For example, to limit a slope in the x-direction, a limiter function should be applied to two differences: Ui,j,k Ui 1,j,k and Ui+1,j,k Ui,j,k. The resulting algorithms ensure the second order of accuracy in smooth regions of the flow. The aim of the present paper is to generalize a TVD principle for a special case of piece-wise uniform grids, i.e., adaptive Cartesian grids. Such grids can be thought of as the result of a multiple refinement procedure applied to an originally uniform Cartesian grid. In the course of the refinement procedure, some of the grid cells are split into two equal parts along each (or some) of the spatial directions (see [3–5], for detail). A particular feature of the adaptive Cartesian grid is the interface that occurs at resolution changes, i.e., refinement interface (RI), through which a coarser cell faces its finer neighbors. Without generalizing the TVD concept for such a configuration of the control volumes, the direct application of the same pair limiter functions as for the uniform grid neither provides second-order accuracy nor ensures the stability of the scheme near the RI. We demonstrate how the TVs from the parts of the grid with different levels of refinement can be merged