Finite Difference Approximation of a Variational Inequality for Fish Migration

摘要

This paper focuses on numerical discretization of a degenerate elliptic variational inequality associated with an optimal control problem that governs fish migration along a 1-D river. Its solutions are discontinuous viscosity solutions that are locally Lipchitz continuous but not necessarily continuously differentiable. A new finite difference scheme with a penalization technique to solve the variational inequality is presented. Its most important key component is utilizing exact viscosity solutions to linear advection-decay equations. Accuracy and stability of the scheme is examined through test cases where exact viscosity solutions are available.