Two Dimensional and Time Varying Spatial Domains

Abstract

Whereas the previous chapters are exclusively dedicated to lumped systems (systems of dimension 0 described by ODEs) and distributed parameter systems in one spatial dimension, this chapter touches upon the important class of problems in more space dimensions, as well as problems with time-varying spatial domains. Both are difficult topics and the ambition of this chapter is just to give a foretaste of possible numerical approaches. Finite difference schemes on simple 2D domains, such as squares, rectangles or more generally convex quadrilaterals, are first introduced, including several examples such as the heat equation, Graetz problem, a tubular chemical reactor, and Burgers equation. Finite element methods, which have more potential than finite difference schemes when considering problems in 2D, are then discussed based on a particular example, namely FitzHugh-Nagumo model. This example also gives the opportunity to apply the proper orthogonal decomposition method to derive reduced-order models. Finally, the problematic of time-varying domains is introduced via another particular application example related to freeze drying. The main idea here is to use a transformation so as to convert the original problem into a conventional one with a time-invariant domain.Whereas the previous chapters are exclusively dedicated to lumped systems (systems of dimension 0 described by ODEs) and distributed parameter systems in one spatial dimension, this chapter touches upon the important class of problems in more space dimensions, as well as problems with time-varying spatial domains. Both are difficult topics and the ambition of this chapter is just to give a foretaste of possible numerical approaches. Finite difference schemes on simple 2D domains, such as squares, rectangles or more generally convex quadrilaterals, are first introduced, including several examples such as the heat equation, Graetz problem, a tubular chemical reactor, and Burgers equation. Finite element methods, which have more potential than finite difference schemes when considering problems in 2D, are then discussed based on a particular example, namely FitzHugh-Nagumo model. This example also gives the opportunity to apply the proper orthogonal decomposition method to derive reduced-order models. Finally, the problematic of time-varying domains is introduced via another particular application example related to freeze drying. The main idea here is to use a transformation so as to convert the original problem into a conventional one with a time-invariant domain.