Basic Concepts and Discretizations
This chapter gives a first introduction to the numerical solution of time-dependent advection-diffusion-reaction problems. Our goal in this chapter is to discuss important basic concepts and discretizations for ordinary differential equations and for adve
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This chapter gives a first introduction to the numerical solution of timedependent advection-diffusion-reaction problems. Our goal in this chapter is to discuss important basic concepts and discretizations for ordinary differential equations and for advection and diffusion equations in one spatial dimension. More advanced material will be treated in later chapters.
1 Advection-Diffusion-Reaction Equations In this first section we will consider some properties of solutions of linear advection and diffusion equations and nonlinear chemical reaction equations and briefly mention some application fields. The standard advection-diffusion-reaction model deals with the time evolution of chemical or biological species in a flowing medium such as water or air. The mathematical equations describing this evolution are partial differential equations (PDEs) that can be derived from mass balances. Consider a concentration u(x, t) of a certain species, with space variable x E lR and time t 2 O. Let h > 0 be a small number, and consider the average concentration u(x, t) in a cell [x - ~h, x + ~hl,
l1X+!h
u(x, t) = -h
x-1.h 2
u(s, t) ds = u(x, t)
+
1 02 _h2 !l 2 u(x, t) 24 uX
+ ....
If the species is carried along by a flowing medium with velocity a(x, t), then the mass conservation law implies that the change of u(x, t) per unit of time is the net balance of inflow and outflow over the cell boundaries,
0_u(x, t) = ot
1] h1[a(x - "21 h, t) u(x - "21 h, t) - a(x + "21 h, t) u(x + "2h, t) ,
where a(x ± ~h, t)u(x ± ~h, t) are the mass fluxes over the left and right cell boundaries. Now, if we let h --t 0, it follows that the concentration satisfies
:tu(x,t) + :x(a(x,t)u(x,t)) =
o.
W. Hundsdorfer et al., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations © Springer-Verlag Berlin Heidelberg 2003
2
I Basic Concepts and Discretizations
This is called an advection equation (or convection equation). 1) In a similar way we can consider the effect of diffusion. Then the change of u(x, t) is caused by gradients in the solution and the fluxes across the cell boundaries are -d(x ± ~h, t)ux(x ± ~h, t) with d(x, t) the diffusion coefficient. The corresponding diffusion equation is
a u(x, t) = ax a ( d(x, t) ax a u(x, t) ) . at There may also be a local change in u(x, t) due to sources, sinks and chemical reactions, which is described by
a
atu(x,t) = f(x,t,u(x,t)). The overall change in concentration is described by combining these three effects, leading to the advection-diffusion-reaction equation
!
u(x, t)
+
:x (a(x, t) u(x, t))
= !(d(x,t):xu(x,t))
(1.1)
+ f(x,t,u(x,t)).
We will consider (1.1) in a spatial interval n c lR with time t > O. An initial profile u(x, 0) will be given and we also assume that suitable boundary conditions are provided. In Section 1.3 the extension to more spatial dimensions is discussed. In the notation we will usually omit the explicit dependence of x and t, and partial derivatives will be denoted by sub-indices. Thus (1.1) will be written as (1.2) Occasionally we w
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