A grid redistribution method for singular problems

Many physical phenomena develop singular, or nearly singular behavior in localized regions, e.g. boundary layers or blowup solutions. Using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity. Ren and Wang developed a semi-static adaptive grid method [W. Ren, X.P. Wang, An iterative grid redistribution method for singular problems in multiple dimensions, J. Comput. Phys. 159 (2000) 246–273] for the solution of these problems, known as the iterative grid redistribution (IGR) method. In this study we develop a theoretical basis for semi-static adaptive grid method for singular problems. Based on this theory, we obtain the key result of this study – a methodology for designing robust weight functionals which ensures grid resolution in the singular region, as well as control of the maximal grid spacing in the outer region. Using this methodology, we introduce a semi-static adaptive grid method, which does not involve an iterative procedure for grid redistribution, as in the IGR method. We demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude.