TY - JOUR
T1 - Approaches to the numerical estimates of grid convergence of NSE in the presence of singularities
AU - Zhang, Chenguang
AU - Nandakumar, Krishnaswamy
N1 - Publisher Copyright:
© 2018 Walter de Gruyter GmbH, Berlin/Boston.
PY - 2018/6/26
Y1 - 2018/6/26
N2 - Evaluating the order of accuracy (order) is an integral part of the development and application of numerical algorithms. Apart from theoretical functional analysis to place bounds on error estimates, numerical experiments are often essential for nonlinear problems to validate the estimates in a reliable answer. The common workflow is to apply the algorithm using successively finer temporal/spatial grid resolutions δi, measure the error ∈i in each solution against the exact solution, the order is then obtained as the slope of the line that fits (log ∈i, log δi). We show that if the problem has singularities like divergence to infinity or discontinuous jump, this common workflow underestimates the order if solution at regions around the singularity is used. Several numerical examples with different levels of complexity are explored. A simple one-dimensional theoretical model shows it is impossible to numerically evaluate the order close to singularity on uniform grids.
AB - Evaluating the order of accuracy (order) is an integral part of the development and application of numerical algorithms. Apart from theoretical functional analysis to place bounds on error estimates, numerical experiments are often essential for nonlinear problems to validate the estimates in a reliable answer. The common workflow is to apply the algorithm using successively finer temporal/spatial grid resolutions δi, measure the error ∈i in each solution against the exact solution, the order is then obtained as the slope of the line that fits (log ∈i, log δi). We show that if the problem has singularities like divergence to infinity or discontinuous jump, this common workflow underestimates the order if solution at regions around the singularity is used. Several numerical examples with different levels of complexity are explored. A simple one-dimensional theoretical model shows it is impossible to numerically evaluate the order close to singularity on uniform grids.
KW - finite volume method
KW - impact of singularity
KW - numerical error estimates
UR - http://www.scopus.com/inward/record.url?scp=85048083794&partnerID=8YFLogxK
U2 - 10.1515/ijnsns-2017-0016
DO - 10.1515/ijnsns-2017-0016
M3 - 文章
AN - SCOPUS:85048083794
SN - 1565-1339
VL - 19
SP - 281
EP - 288
JO - International Journal of Nonlinear Sciences and Numerical Simulation
JF - International Journal of Nonlinear Sciences and Numerical Simulation
IS - 3-4
ER -