TY - JOUR
T1 - A grid redistribution method for singular problems
AU - Ditkowski, Adi
AU - Gavish, Nir
N1 - Funding Information:
Special thanks to Xiao-Ping Wang who introduced us to this subject, for sharing his insights and for his guidance in the implementation of the IGR method. We thank Gadi Fibich and Yonatan Sivan for useful discussions. The research of Adi Ditkowski was supported by the Israel Science Foundation (grant no. 1364/04), and the United States–Israel Binational Science Foundation (grant no. 2004-099). The research of Nir Gavish was partially supported by grant number 123/08 from the Israel Science Foundation (ISF) and by the Israel Ministry of Science Culture and Sports.
PY - 2009/4/20
Y1 - 2009/4/20
N2 - 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.
AB - 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.
KW - Grid generation
KW - Iterative grid redistribution (IGR)
KW - Nonlinear Schrödinger (NLS)
KW - Singular problems
UR - http://www.scopus.com/inward/record.url?scp=60149097709&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2008.11.035
DO - 10.1016/j.jcp.2008.11.035
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AN - SCOPUS:60149097709
SN - 0021-9991
VL - 228
SP - 2354
EP - 2365
JO - Journal of Computational Physics
JF - Journal of Computational Physics
IS - 7
ER -