Seite - 288 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

288 9 SolvingPartialDifferentialEquations theequationisknownastheheatequation.Weremarkthat thetemperatureinafluid is influenced not only by diffusion, but also by the flow of the liquid. If present, the latter effect requires an extra term in the equation (known as an advection or convectionterm). Thetermg isknownasthesourcetermandrepresentsgeneration,orloss,ofheat (by some mechanism) within the body. For diffusive transport,g models injection orextractionof thesubstance. Weshouldalsomention that thediffusionequationmayappearafter simplifying morecomplicatedPDEs. For example,flow of a viscousfluid between two flat and parallel plates is described by a one-dimensional diffusion equation, whereu then is thefluidvelocity. A PDE is solved in some domainΩ in space and for a time interval [0,T]. The solutionof theequation isnotuniqueunlesswealso prescribe initialandboundary conditions.Thetypeandnumberofsuchconditionsdependonthe typeofequation. For the diffusion equation, we need one initial condition,u(x,0), stating what u is when the process starts. In addition, the diffusion equation needs one boundary conditionat each point of the boundary∂Ω ofΩ. This conditioncan either be that u is known or that we know the normal derivative,∇u ·n= ∂u/∂n (ndenotes an outwardunitnormal to∂Ω). 9.1 Example:TemperatureDevelopmentinaRod Let us look at a specific application and how the diffusion equation with initial and boundary conditions then appears. We consider the evolution of temperature in a one-dimensional medium, more precisely a long rod, where the surface of the rod is covered by an insulating material. The heat can then not escape from the surface, which means that the temperature distribution will only depend on a coordinate along the rod, x, and time t. At one end of the rod, x = L, we also assume that the surface is insulated, but at the other end, x = 0, we assume that we have some device for controlling the temperature of the medium. Here, a function s(t) tells what the temperature is in time. We therefore have a boundary condition u(0,t) = s(t). At the other insulated end, x = L, heat cannot escape, which is expressed by the boundary condition ∂u(L,t)/∂x = 0. The surface along the rod is also insulated and hence subject to the same boundary condition (here generalized to ∂u/∂n = 0 at the curved surface). However, since we have reduced the problem to one dimension, we do not need this physical boundary condition in our mathematical model. In one dimension, we can set Ω=[0,L]. Tosummarize, thePDE with initial andboundaryconditionsreads ∂u(x,t) ∂t =β∂ 2u(x,t) ∂x2 +g(x,t), x∈ (0,L),t∈ (0,T], (9.1) u(0,t)= s(t), t∈ (0,T], (9.2)