Volume 13, Issue 3
A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 620-643.

Published online: 2020-03

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• Abstract

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.

• Keywords

Numerical methods, nonlinear elliptic equations, obstacle problem, semi-implicit scheme.

65N06, 65N12, 35J60

hao.liu@math.gatech.edu (Hao Liu)

masyleung@ust.hk (Shingyu Leung)

• BibTex
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@Article{NMTMA-13-620, author = {Liu , Hao and Leung , Shingyu}, title = {A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {3}, pages = {620--643}, abstract = {

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0126}, url = {http://global-sci.org/intro/article_detail/nmtma/15778.html} }
TY - JOUR T1 - A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints AU - Liu , Hao AU - Leung , Shingyu JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 620 EP - 643 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0126 UR - https://global-sci.org/intro/article_detail/nmtma/15778.html KW - Numerical methods, nonlinear elliptic equations, obstacle problem, semi-implicit scheme. AB -

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.

Hao Liu & Shingyu Leung. (2020). A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints. Numerical Mathematics: Theory, Methods and Applications. 13 (3). 620-643. doi:10.4208/nmtma.OA-2019-0126
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