1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 25

2 Resuming Proof of Convergence

3 2D Advection Equation Equation: Scheme (assuming s’th order time integration):

4 Summary of Needed Theorems: We will now use the

5 Convergence Theorem Assume that a solution exists in the domain Omega. Then the numerical solution Ctilde to the semidiscrete approximation from the DG scheme converges to the exact solution, and the global error is bounded as: Where c (lower case) is independent of h and p.

6 Proof Step 1: Recall the DG scheme: We define the Truncation error T C as

7 Proof Step 2:Three Important Equations We now consider three equations: 1)Numerical: 2)Truncation – projection of exact solution in numerical scheme: 3)Exact equation, with exact solution:

8 Proof Step 4: Truncation Error Subtract equation 3 from equation 2: Substitute phi=Tc and apply Cauchy-Schwarz:

9 Proof Step 4: cont Substituting phi=T c

10 Individual Volume Terms: Assume exact time discretization: Using the polynomial approximation theorem:

11 Surface Term We can add the jump in C due to continuity of C:

12 Jump Terms Break the jump terms with triangle inequality and use the trace theorem:

13 Plugging In These Results Using the previous results and the polynomial inverse trace inequality:

14 Simplifying We can now bound the truncation error: We now know the price we are paying at every time step for using a finite, discrete space. The estimates are constructed using the L2- projection and norm. The estimates can be improved using a more sophisticated norm and projection.

15 Now The Conclusion We now need to track how far the numerical solution compares to the L2 projection of the actual solution diverge in time. Then we find out how far the L2 projection of the actual solution can be from the actual solution at a given time.

16 Recall Equations 1 & 2 1)DG scheme 2)DG scheme acting on projection of solution and generated truncation term:

17 Difference of Equations 1 & 2 Subtracting (2) from (1):

18 Estimating Since both numerical solution Ctilde and the projection of the exact solution is in the p’th order space:

19 Simplify Taking the time derivative out of the inner-product:

20 Look at Total Gap We now sum over all elements to look for the total gap between the numerical and projected solution:

21 Use Stability We should now recognize that the sum of the first two terms on the right hand side is negative (here we can assume that the boundary contributions are zero as the numerical solution and projection of C should agree at the boundary):

22 Bounding We next use Cauchy-Schwarz:

23 Bringing It All Together: To obtain the red terms we plugged in estimate for the truncation error To obtain the blue term we plugged in the estimate for the projection of the initial condition

24 End Game We now know how far away the DG solution is from the L2 projection of the exact solution: So one last step we estimate the total error as:

25 Finally: So let’s recap where each term comes from:

26 Finally: So let’s recap where each term comes from: L2 norm of total error Initial projection error Accumulation of truncation error in time. Portion of solution outside polynomial space

27 Comment Ok – we now know that if the solution we are trying to compute has enough regularity then the error is going to be of order : sigma for small time t sigma-1 after longer time

28 Enough Theory I am going to provide you with a basic code which solves the advection equation with the DG scheme Keep in mind that we will soon be converting this code to work for other 2D PDEs (Maxwell’s, Acoustics..) Your job is to compute the interpolation nodes on the triangle using the code from umFEKETE and run the code on some test cases.

29 Homework 8 1) Grab copy of umSCALAR2d.zip from cd-rom 2) Unzip the file 3) Use the umFEKETE codes to create the nodes for p=1,…,10 and save into one file for each order. 4) Edit umSCALAR2d/umSCALARdemo.m to load the node locations into the umR and umS vectors for the given polynomial order. 5) Run the umSCALARdemo in Matlab 6) Plot p-convergence (choosing a final time when the pulse is still inside the domain) 7) Create your own sequence of refined meshes with the windows code for say a square domain. 8) For p=1,…,10 model the error as h^order and estimate the value of order.

30 Here’s the code you modify in umSCALARdemo.m

31 Nect lecture Time to finish the homework – but make sure you make a good stab at this before hand I will collect your efforts at the end of class!!!.