TY - JOUR
T1 - Fast operator splitting methods for obstacle problems
AU - Liu, Hao
AU - Wang, Dong
N1 - Funding Information:
H. Liu acknowledges support from National Natural Science Foundation of China grant (Grant No. 12201530 ) and Hong Kong Baptist University (Grant No. 179356 and 162784 ). D. Wang acknowledges support from National Natural Science Foundation of China grant (Grant No. 12101524 ), Shenzhen Science and Technology Innovation Program (Grant No. JCYJ20220530143803007 ) and the University Development Fund from The Chinese University of Hong Kong, Shenzhen ( UDF01001803 ).
Publisher Copyright:
© 2023 Elsevier Inc.
PY - 2023/3/15
Y1 - 2023/3/15
N2 - The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator–splitting methods to solve the linear and nonlinear obstacle problems. The proposed methods have three ingredients: (i) Utilize an indicator function to formularize the constrained problem as an unconstrained problem, and associate it to an initial value problem. The obstacle problem is then converted to solving for the steady state solution of an initial value problem. (ii) An operator–splitting strategy to time discretize the initial value problem. After splitting, a heat equation with obstacles is solved and other subproblems either have explicit solutions or can be solved efficiently. (iii) A new constrained alternating direction explicit method, a fully explicit method, to solve heat equations with obstacles. The proposed methods are easy to implement, do not require to solve any linear systems and are more efficient than existing numerical methods while keeping similar accuracy. Extensions of the proposed methods to related free boundary problems are also discussed.
AB - The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator–splitting methods to solve the linear and nonlinear obstacle problems. The proposed methods have three ingredients: (i) Utilize an indicator function to formularize the constrained problem as an unconstrained problem, and associate it to an initial value problem. The obstacle problem is then converted to solving for the steady state solution of an initial value problem. (ii) An operator–splitting strategy to time discretize the initial value problem. After splitting, a heat equation with obstacles is solved and other subproblems either have explicit solutions or can be solved efficiently. (iii) A new constrained alternating direction explicit method, a fully explicit method, to solve heat equations with obstacles. The proposed methods are easy to implement, do not require to solve any linear systems and are more efficient than existing numerical methods while keeping similar accuracy. Extensions of the proposed methods to related free boundary problems are also discussed.
KW - Obstacle problem
KW - Operator splitting methods
KW - Alternating direction explicit method
KW - Nonlinear elliptic equations
KW - Free boundary
UR - http://www.scopus.com/inward/record.url?scp=85146865388&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2023.111941
DO - 10.1016/j.jcp.2023.111941
M3 - Journal article
AN - SCOPUS:85146865388
SN - 0021-9991
VL - 477
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 111941
ER -