The essentially nonoscillatory (ENO) and weighted essentially nonoscillatory (WENO) schemes are methods used to represent the meshes used to solve partial differential equations when there is an abrupt change in conditions governing the behavior of the system. The partition of the mesh adapts to the problem in a manner that prevents oscillatory behavior at the sudden change. The oscillatory behavior cannot be damped, and finding solutions can be frustrating. The challenge is to find suitable procedures for adapting the mesh stencil to the problem so that the calculation can proceed accurately and efficiently.
In this paper, the authors present a method for adapting the mesh based on the quality of the solution being developed by measuring the discrepancy between the projected ENO and WENO solutions in the region of the discontinuity. This allows the mesh stencil to move with the solution as it evolves. They also investigate strategies for optimizing the efficiency of the process, such as deferring changes to the mesh until after a few iterations versus after every iteration.
The paper provides an exposition of the WENO scheme on nonuniform meshes, methods for moving meshes, the algorithm, and several numerical examples. The narrative section with the greatest detail is on methods for moving meshes. There are five issues that must be dealt with in moving a mesh: estimating the error, the choice of strategy for moving the mesh, estimation of the number of new cells that must be introduced (and reduction of the count of cells where the discontinuity is now absent), interpolation on the new mesh, and integration of the physical equations. The algorithm is presented in six steps with detailed notes about performance and numerical constraints that must be adhered to.
There are four classes of problems providing detailed examples: linear advection, Burgers’ equation, Euler equations, and Lax’s and Sod’s shock tube problems. The examples are richly illustrated with graphs showing the quality of the solutions obtained.
