A stability analysis of a Eulerian solution method for moving boundary problems in electrochemical machining

摘要

We investigate the front-tracking method using marker points for the numerical solution of a class of moving boundary problems, which arise in connection with certain two-dimensional electrochemical machining problems. The numerical method is expressed as a system of N nonlinear, autonomous, ordinary differential equations (ODEs) for describing the movement of N marker points on the moving boundary. The system has been analysed, in some cases, with respect to the local stability of equilibrium solutions and to the global stability of nonequilibrium solutions. The linearised local stability of the equilibrium solution is investigated in terms of the Jacobi matrix for the system of ODES. Closed-form expressions for the eigenvalues and the eigenvectors of the Jacobi matrix have been derived. Part of the method comprises the solution of a Dirichlet boundary value problem. This is carried out using various boundary collocation methods, which are investigated. The mathematical-numerical model has features which are in accordance with those of the physical problem. This has also been demonstrated by actual computation of some examples.