Invited Lecture
High-order IMEX and semi-implicit multistep methods for evolutive partial differential equations
University of Verona, Italy
Abstract
In this talk we will revise the construction of different classes of semi-implicit multi-step method for time-dependent PDEs. First of all we will consider the development of high-order space, and time numerical methods based on implicit-explicit multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. Hence, we design implicit-explicit linear multistep methods at high-order space-time discretization which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. We will also consider the construction of semi-implicit linear multistep methods which can be applied to time dependent PDEs where the separation of scales in additive form, is not possible. These semi-implicit techniques give a great flexibility, and allows, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. We will discuss both set-up on different numerical examples, including nonlinear reaction-diffusion and convection-diffusion problems.