A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

Hao Liu*, Shingyu Leung

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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.
Original languageEnglish
Pages (from-to)620-643
Number of pages24
JournalNumerical Mathematics
Volume13
Issue number3
DOIs
Publication statusPublished - Aug 2020

Scopus Subject Areas

  • Modelling and Simulation
  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Nonlinear elliptic equations
  • Numerical methods
  • Obstacle problem
  • Semi-implicit scheme

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