A multilevel framework for PDEs whose solution exhibits fast transitions.

摘要

We introduce an adaptive multilevel framework for the solution of numerical partial differential equations (PDEs) whose solution exhibits codimension-one discontinuities, or fast transitions. Our framework has three main components: grid generation, derivative evaluation and solution integration. The grid generation portion is based on a linear version of Harten's generalized multiresolution analysis (MRA), which we use to bound discontinuities in R . We then extend this MRA to Rn in terms of n-orthogonal linear transforms to identify cells that contain a codimension-one discontinuity. These refinement cells become leaf nodes in a balanced kD-tree such that a local dyadic MRA is produced in Rn , while maintaining a minimal computational footprint. The nodes in the tree form an adaptive mesh whose density increases in the vicinity of a fast transition.;Utilizing the multilevel information encoded in the kD-tree, we developed a multilevel multiquadric radial basis function (RBF) that is scale-aware. These multilevel RBFs can interpolate nodal values between different kD-trees without generating Gibbs' effects near a codimension-one discontinuity. This interpolation technique was extended to form a scale-aware RBF differential quadrature method that can evaluate derivatives on the kD-trees. Our differential quadrature method is capable of representing derivatives of the sampled solution surface on balanced kD-trees without generating Gibbs' effects near codimension-one discontinuities, supposing there is some minimal separation distance between each fast transition. In addition to the grid generation and derivative portion of our framework, we detail our ongoing research on the adaptive multilevel integration, and show some preliminary results.