Backward Error Compensation for Transportation Equations with Application to the Level Set Method
Yingjie Liu - Georgia Institute of Technology

Level set method uses a level set function, usually an approximate signed distance function, Phi, to represent the interface as the zero set of Phi. When Phi is advanced to the next time level by a transportation equation, its new zero level set will represent the new interface position. We update the level set function Phi forward in time and then backward to get another copy of the level set function, say Phi_1. Phi_1 and Phi should have been equal if there were no numerical error. Therefore Phi-Phi_1 provides us the information of error and this information can be used to compensate Phi before updating Phi forward again in time. One nice property is that it has the convenience of possibly improving the temporal and spatial order of an odd order scheme simultaneously. We found that when applying this idea to semi-Lagrangian schemes, e.g., CIR scheme(which has no CFL restriction, a nice feature for local refinement), the property is still valid (while MacCormack scheme having similar property may not be easily applied here). Numerical results for interface movements with level set equation computed by the new methods will be presented in the talk. Also I would like show some interesting theoretical results for applying this idea to a general linear scheme.