ACCURATE FINITE-DIFFERENCE APPROXIMATIONS FOR THE NEUMANN CONDITION ON A CURVED BOUNDARY

When finite difference techniques are used at curved boundaries, a common treatment is to reshape the boundary so that it is made up of straight line segments that coincide with grid lines. This treatment greatly decreases the accuracy of the finite difference approximations at points near the boundary. In this paper we develop an accurate explicit finite difference scheme for approximating the derivatives at points that are near a curved boundary along which a Neumann boundary condition applies. The curved boundary does not need to be reshaped, and the approximations applied at adjacent grid points have the same accuracy as schemes commonly used for Dirichlet boundary conditions. Numerical tests involving wind effects on a circular lake show that the performance of the new method is much superior to the conventional approach of boundary reshaping. © 1990.