TY - JOUR
T1 - An adaptive multiresolution discontinuous galerkin method for time-dependent transport equations in multidimensions
AU - Guo, Wei
AU - Cheng, Yingda
N1 - Funding Information:
∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section July 5, 2016; accepted for publication (in revised form) June 2, 2017; published electronically December 14, 2017. http://www.siam.org/journals/sisc/39-6/M108319.html Funding: The first author’s research was supported by NSF grant DMS-1620047. The second author’s research was supported by NSF grant DMS-1453661. †Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 70409 (weimath.guo@ttu.edu). ‡Department of Mathematics, Michigan State University, East Lansing, MI 48824 (ycheng@math.msu.edu).
Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.
PY - 2017
Y1 - 2017
N2 - In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for time-dependent transport equations in multidimensions. The method is constructed using multiwavlelets on tensorized nested grids. Adaptivity is realized by error thresholding based on the hierarchical surplus, and the Runge–Kutta DG scheme is employed as the reference time evolution algorithm. We show that the scheme performs similarly to a sparse grid DG method when the solution is smooth, reducing computational cost in multidimensions. When the solution is no longer smooth, the adaptive algorithm can automatically capture fine local structures. The method is therefore very suitable for deterministic kinetic simulations. Numerical results including several benchmark tests, the Vlasov–Poisson (VP), and oscillatory VP systems are provided.
AB - In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for time-dependent transport equations in multidimensions. The method is constructed using multiwavlelets on tensorized nested grids. Adaptivity is realized by error thresholding based on the hierarchical surplus, and the Runge–Kutta DG scheme is employed as the reference time evolution algorithm. We show that the scheme performs similarly to a sparse grid DG method when the solution is smooth, reducing computational cost in multidimensions. When the solution is no longer smooth, the adaptive algorithm can automatically capture fine local structures. The method is therefore very suitable for deterministic kinetic simulations. Numerical results including several benchmark tests, the Vlasov–Poisson (VP), and oscillatory VP systems are provided.
KW - Adaptive multiresolution analysis
KW - Discontinuous Galerkin methods
KW - Sparse grids
KW - Transport equations
KW - Vlasov–Poisson system
UR - http://www.scopus.com/inward/record.url?scp=85040009209&partnerID=8YFLogxK
U2 - 10.1137/16M1083190
DO - 10.1137/16M1083190
M3 - Article
AN - SCOPUS:85040009209
VL - 39
SP - A2962-A2992
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
SN - 1064-8275
IS - 6
ER -