Essentially Non-Oscillatory Adaptive Tree Methods

We develop high order essentially non-oscillatory (ENO) schemes on non-uniform meshes based on generalized binary trees. The idea is to adopt an appropriate data structure which allows to communicate information easily between unstructured data structure and virtual uniform meshes. While the generalized binary trees as an unstructured data structure can store solution information efficiently if combined with a good adaptive strategy, virtual uniform meshes allow us to take advantage of many well-developed ENO numerical methods based on uniform meshes. Therefore, the ENO adaptive tree methods proposed here can leverage the merits from both tree structures and uniform meshes. Numerical examples demonstrate that the new method is efficient and accurate.