Convection - diffusion problem

FINITE ELEMENT METHODS IN FLUIDS
Master of Science in Computational Mechanics – A.Y. 2011
ASSIGNMENT #2: Unsteady Convection-Diffusion Problems
Due date: May 4, 2011

Natalia Valencia Arango

Exercise 1
Consider the transport problem: ut+a∙∇u=0 in -12,12×-12,12

With homogeneous Dirichlet boundary conditions on the inlet boundary and initial condition:ux,y,0=14(1+cosπX)(1+cosπY) if X2+Y2≤10 Otherwise

Where X,Y=x-16σ,y-16σ, σ=0.2, a=(-y,x)

A. To solve this problem, we want to use the Lax-Wendroff, the Crank-Nicolson and the third-order explicit Taylor-Galerkin (TG3) methods. Write the formulation of these methods to solve the given problem.

The main feature of this equation is that space and time are linked by the characteristics. To compute a numericalsolution, a double discretization has to be performed (in space and in time) and to obtain accurate solutions it cannot be done anyhow.

Stability of numerical methods depends on the Courant number:C=a∆th, That links spatial and time discretization.

It is also important the order in which discretizations are performed. We can see the discretization formulations for each method:

Lax-Wendroff:Explicit, second order accuracy

+ Galerkin

Crank-Nicolson: Implicit, second order accuracy, unconditionally stable.

+ Galerkin

Third-order explicit Taylor-Galerkin (TG3): Higher order

+ Galerkin

Now, we have to take into account the stability analysis that is done fallowing three steps:
* Fourier series of the initial condition is considered
ux,0=n=-∞∞Cneiknx
*Linearity assumption: the evolution of the initial condition is the sum of the evolutions for each single mode,eikx
* The exact and numerical amplification factors are computed un+1=Gun relates the unknown at two consecutive instants. We have to take into account:

The Courant number: C=a∆th
The diffusion number: d=ν∆th2
The reaction number: r=σΔt

G is the factor to compute, it is importantfor the stability that the value of G≤1.
For each method we have a different formulation of the amplification factor:

Lax-Wendroff: Explicit, second order accuracy

Crank-Nicolson: Implicit, second order accuracy, unconditionally stable.

Third-order explicit Taylor-Galerkin (TG3): Higher order
For this method the formulations are similar to the second order Lax-Wendroff but we use a Taylorseries expansion up to order 3 (it provides a third order accuracy).

B. Compute the solution at t = 2π using the Lax-Wendroff method with consistent mass matrix. Use a mesh of 20×20 bilinear elements and 110 and 120 time steps. Comment on the differences between the two computed solutions.

When we use a time step equal to 110, we don’t have a numerical solution, that is because thesystem is not stable and we need more time steps in order to have a solution. In the code you can see in the main in the postprocess part, that there is a condition that say: if the max(u(:,nstep+1))<100 and the min(u(:,nstep+1))>100 you do the postprocess but if this condition is not ok, you don’t do the postprocess, that is why you don’t have a solution.

If we erase that condition we can seede solution that is not stable.



Ilustración 1 Lax Wendroff, 20x20, TS 110, CM
When we use a time step equal to 120 we have the numerical solution in the figure 2, now there are convergence, the system is stable and we have a good solution.

Ilustración [ 2 ] Lax Wendroff, 20x20, TS 120, CM
Repeat the computations using the Lax-Wendroff method with diagonal mass matrix. Compare thesesolutions with the ones obtained before, paying special attention to the stability and accuracy of the numerical solutions.

Using the diagonal mass matrix we have the solution in figure 3 and 4, first of all we can see that in this case we have a numerical solution with time step equal to 110 (fig 3), that means that with the diagonal mass matrix we need a time step smaller than in the...