1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 17

2 Stability Regions For s=1,2,3,4 Eigenspectrum for DG (N=6, p=3) Matrix for DG spatial operator dt*spectrum must fit in the closed loop for a given s s=1 s=2 s=3 s=4

3 Advection-Diffusion Equation We now generalize the advection equation to include possible diffusion effects. Examples of diffusion driven processes ??

4 Advection-Diffusion Equation We consider the motion of a tracer solute in a fluid with mean velocity ubar. We will denote the concentration of the tracer by C(x,t) Not only is the tracer advected by the fluid’s mean velocity – it will also diffuse by random particle motion. Ubar is typically the mean velocity of the fluid particles – actual particle velocity deviates randomly from this. The advection-diffusion equation is:

5 Advection The advection equation is as previously discussed: i.e. the total tracer in a section of pipe is only changed by advection of the tracer through the ends of the section. However, if the tracer particles are moving randomly then particles will randomly jump in and out of the section of pipe.

6 Diffusion An underlying assumption of the ADE is that mechanical dispersion, like molecular diffusion, can be described by Fick’s first law: where F is the mass flux of solute per unit area per unit time and D is the effective diffusion coefficient in a porous medium. Fick.s law states that particle flux is directly proportional to the spatial concentration gradient. But it is not the spatial concentration gradient that causes particle movement, i.e. particles do not.push. each other (Crank, 1976). Particles exhibit random motion on the molecular level. This random motion ensures that a tracer will diffuse, decreasing the concentration gradient (Crank, 1976). Crank, J., The Mathematics of Diffusion. Oxford University Press, New York.

7 Diagram of Diffusion Model a b b Assume that each particle is jumping with a rate of R(delta) jumps per second which take it a distance of delta or more then there will be a number of jumps out of the left delta width: We count the number of jumps in from the right delta width: Summing:

8 Quick Derivation of Fickian Diffusion Consider the x=b end of the section. We are going to “monitor” the random motions of a particle in and out of the region: Assume that each particle is jumping with a rate of R jumps in the +ve direction of (length>delta) per second then there will be a flux of out of the b end (similar at the a end) We apply tracer counting (and approximation):

9 We now recall that R is a function of delta and clearly R must be inversely proportional to delta^2. i.e. as the region we are monitoring shrinks to zero, the rate of random motions into and out of the control region increases… We denote and obtain: [ Note continuity assumptions ]

10 DG Scheme for the Scalar ADE (Lax-Friedrichs flux – equivalent to upwind)

11 DG Derivative Operator We are going to introduce a DG derivative operator to simplify the scheme definition. The linear operator Dtilde is such that the following holds for all intervals Ij With the choice of penalty terms tauL and tauR to be determined.

12 In Operator Notation Advection term ( Lax-Friedrichs flux ~ upwind flux for scalar case ) Diffusion term

13 D operator If we move to the Legendre basis we can write down a discrete description of Dtilde:

14 Discrete Scheme Step 1: q = x DG derivative of C Step 2: time rate of change of C Advection term ( Lax-Friedrichs flux ~ upwind flux for scalar case ) Diffusion term

15 Dropping q We can do away with q: This should make the following observation clear. The dt dependence for stability is now: The first term is due to the spectral radius of the advection operator. The second term is due to the spectral radius of the diffusion operator.

16 Next Time Details on applying Dirichlet and/or Neumann boundary conditions in the given formulation.

17 Project Time Continue working in pairs. Recall – the project is due at the beginning of class on Monday 03/03/03 Recall – presentation due on Monday, make sure you are ready. Recall – late presentations and projects will be penalized by 20% per day.