On some oscillatory properties of finite difference methods for one-dimensional nonlinear parabolic problems

Abstract: In this talk, we will discuss some qualitative properties of numerical solutions of parabolic partial differential equations. First, we will use some examples to show the importance of the preservation of these properties. Then we will consider two special properties of the finite difference solutions of nonlinear problems in detail: the first property says that the number of the sign-changes of the solution function must be non-increasing in time; the second property requires a similar property for the number of the local maximizers and minimizers. We recall and formulate some theorems that guarantee the above properties for the solution of the continuous problem. We generate the numerical solution and give sufficient conditions for the mesh size and the time step that guarantee the discrete versions of the properties. We also present some numerical test results.