A Finite-Difference Logical-Coordinate Evaluation Method for Particle Localization

Abstract

Particle methods require robust and efficient advection and localization methods which include logical-coordinate evaluation. The ability to compute logical coordinates with existing methods is, however, not guaranteed within grids that contain nonlinear elements. This note presents a new logical-coordinate evaluation method, based on finite-differences, that provides a robust and efficient solution for coordinate transformation. The new methods enhancedcapabilities are d emonstratedon a simple test problem. Particle methods, computational models of particle dynamics, require robust and efficient advection and localization methods. Localization methods 1) - 6) combine cell-searching and logical-coordinate evaluation methods to define particle-grid connectivity. This connectivity data consist of the identity of the grid cell in which the particle resides and the particle’s position relative to that cell, its transformed or logical coordinates. Cell-searching or guessing methods typically use the particle’s logical coordinates to both direct and halt the search. Particle methods are, therefore, predicated on robust and efficient logical-coordinate evaluation methods. Existing logical-coordinate evaluation methods are generalized in Ref. 6). Solutions usingthis technique are g uaranteed, and the coordinate vector is bound between known transformation limits, if the particle resides within the guessed cell. In contrast, an unbound coordinate solution may fail to exist for nonlinear grid element transformations. The problem of interest, however, occurs when these coordinates are unbound because only then will the particle have exited the cell duringadvection. This note continues by presentinga new evaluation method that is less sensitive to coordinate transformation. A test problem concludes this note. §2. Finite-difference evaluation method