1. 1 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu MA557/MA578/CS557 Lecture 19

2. 2 Advection-Diffusion Equation

3. 3 DG Scheme for the Scalar ADE (Lax-Friedrichs flux – equivalent to upwind)

4. 4 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.

5. 5 In Operator Notation Advection term ( Lax-Friedrichs flux ~ upwind flux for scalar case ) Diffusion term

6. 6 General Dtilde Operator We previously defined Dtilde for the special case where the test function is compactly supported on the j’th cell. We generalize by:

7. 7 Skew-symmetry of Dtilde 1,1 [integrate by parts]

8. 8 Skew-symmetry Note: the negative sign when the operator moves. Trivially:

9. 9 Recap We already proved that the DG scheme, with Lax-Friedrichs fluxes, for advection (D=0) is stable in the sense that : Adding the diffusion term (dropping boundary terms) gives: In terms of norms:

10. 10 Stability!! Assuming D>=0 (reasonable since particles do not jump randomly with negative rates)

11. 11 D operator If we move to the Legendre basis we can write down a discrete description of Dtilde:

12. 12 Implementation of Dtilde function [dfdx] = LEGdgderiv(f, nodex, tauL, tauR, D, F, G, H, J) [p,N] = size(f); % p = maximum polynomial order p=p-1; % N = number of nodes N=N+1; % space for derivative dfdx = zeros(p+1,N-1); % cells 2 to N-2 ids = (2:N-2); dfdx(:,ids) = (D*f(:,ids) + tauL*F*f(:,ids) + tauL*G*f(:,ids-1) + tauR*H*f(:,ids) + tauR*J*f(:,ids+1)); % cell 1 (periodicity assumed) dfdx(:,1) = (D*f(:,1) + tauL*F*f(:,1) + tauL*G*f(:,N-1) + tauR*H*f(:,1) + tauR*J*f(:,2)); % cell N-1 (periodicity assumed) dfdx(:,N-1) = (D*f(:,N-1) + tauL*F*f(:,N-1) + tauL*G*f(:,N-2) + tauR*H*f(:,N-1) + tauR*J*f(:,1)); % apply chain rule for physical cell width dx = nodex(2:N)-nodex(1:N-1); coeff = ones(p+1,1)*(2./dx); dfdx = dfdx.*coeff;

13. 13 In Action (advection eqn) % TIME STEPPING tauL = (1/u)*(u+abs(u))/2; tauR = (1/u)*(u-abs(u))/2; for tstep = 1:Ntsteps sigma = rho; for rkstage = 3:-1:1 dsigmadx = LEGdgderiv(sigma, nodex, tauL, tauR, D, F, G, H, J); sigma = rho + (dt/rkstage)*(-u*dsigmadx); end rho = sigma; if ( ~mod(tstep, 100) ) q = V*rho; % plot(x(:),abs(q(:)-exp(-(x(:)-dt*tstep).^2))); plot(x,q); pause(0.1); tstep*dt end

14. 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. 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. 16 In Action (ADE) % TIME STEPPING for tstep = 1:Ntsteps sigma = rho; for rkstage = p:-1:1 % advection: upwind bias on flux terms dsigmadx = LEGdgderiv(sigma, nodex, tauL, tauR, D, F, G, H, J); % diffusion: centered fluxes dsdx = LEGdgderiv(sigma, nodex, 1, 1, D, F, G, H, J); d2sdx2 = LEGdgderiv(dsdx, nodex, 1, 1, D, F, G, H, J); sigma = rho + (dt/rkstage)*(-u*dsigmadx+Dcoeff*d2sdx2); end rho = sigma; if ( ~mod(tstep, 400) ) q = V*rho; plot(x,q); axis([xmin xmax 0 1]) pause(0.1); end

17. 17 Downloads You can download the Dtilde based scripts from: http://www.math.unm.edu/~timwar/MA578S03/MatlabScripts/LEGadvectionLaxF.m http://www.math.unm.edu/~timwar/MA578S03/MatlabScripts/LEGade.m

18. 18 Legendre Scheme To Lagrange Scheme (modal to nodal) So far we have used Legendre expansions to represent the solution in each cell. This representation causes some problems if we wish to evaluate non-linear function of the solution. For instance solving the inviscid Burger’s equation: requires the evaluation of u 2

19. 19 Transforming the Modal Advection Scheme Recall that given a Legendre expansion we may evaluate it at a set of points by: We substitute: into the scheme:

20. 20 Nodal Scheme Multiplying both sides on the left with V This clearly shows that the nodal scheme is a change of basis away from the modal scheme.

21. 21 Summary of Nodal Scheme Note: we do not need to transform the initial condition or solution for visualization…

22. 22 Next Lecture 1)The lecture on Friday 03/07 is cancelled in favour of the Analysis Conference 2)The lecture on Monday 03/10 will be the first time we generalize the DG to two spatial dimensions. 3)Do not miss the Monday lecture 4)Run the LEGade.m script (you will need the LEGdgderiv.m script as well) 5)How to use nodal methods in general.