Integro-Interpolation Method of Constructing a Difference Scheme in a Problem with a Moving Boundary

Abstract

Working with systems that involve moving boundaries can be a very difficult task. Not only do we have to solve the equations describing the system, but we also have to find the region the system occupies at each step. One of the common moving-boundary classes, Stefan problems are systems of diffusion or heat-conduction where the boundaries between the different phases in the system change over time [1, 2]. Unfortunately, since Stefan problems can be so complex that an analytical solution of the system is often impossible. Therefore, approximate analytical methods or numerical methods, which are the most practical for working with these problems, are often used.  This work is devoted to numerical investigation of nonlinear fluid filtration. Hydrodynamic study of non-Newtonian fluid filtration requires solving nonlinear differential equations with partial derivatives. The integration of these equations is associated with serious mathematical difficulties caused by moving boundaries, the dependence of the physical properties on the coordinates and time, the specifics of the boundary conditions. Therefore, in the works devoted to the study of nonlinear effects of filtering liquid and gas, approximate methods are used (quasistationary approximation, the integral relations and numerical). Among them, we can note the simplicity and versatility of finite difference method, which, however, requires the solution of a complex system of algebraic equations with simple computational algorithms.  In our problem, in order to close the mathematical system, another equation is required is a type of Stefan's condition. This is the law of conservation of momentum balance, which determines the position of the moving interface. Note that this moving boundary is an unknown surface. Consequently, the problem we are considering is an example of a free boundary problem [3].