Last time: Diffusion - Numerical scheme (FD) Heat equation is dissipative, so why not try Forward Euler:

Transcription

1 Lecture

2 Last time: Diffusion - Numerical scheme (FD) Heat equation is dissipative, so why not try Forward Euler: U j,n+1 t U j,n = U j+1,n 2U j,n + U j 1,n x 2 Expected accuracy: O(Δt) in time, O(Δx 2 ) in space. t Stability in the usual way gives R = x 2 apple 1 2 We can use previous ODE methods using the old method of lines, of course Or implicit methods (see book, 6.5). Notice that matrix contains O(Δx -2 ) terms and is stiff!

3 Advection + diffusion For incompressible flows, a concentration c advects and diffuses via a combination of a heat with 1-way wave equation: 1D: c t = Dc xx + vc x Peclet number measures dominance of diffusion vs. advection: = cl D Remember: Diffusion is easy, but advection (1-way!) is tricky: numerical dispersion, oscillations (e.g. Lax-Wendroff). So, for Pe<<1, it is usually ok to just solve the diffusion problem! What about Pe>>1? Can we do the same (drop diffusion term)?

4 Advection + diffusion For a characteristic scale of L0=D/c: Length scales below L0: Diffusion-dominated Length scales above L0: Advection-dominated L0=D/c=1/40 L0=D/c=1/20 L0=D/c=1/10 Small L0=D/c (i.e. large Peclet number) creates a boundary layer, where the solution significantly deviates from the pure advection solution u(x) = 0 Lecture: Example ~L0

5 Advection + diffusion In other words: To correctly resolve physics at smaller scales L<L0 (in particular inside the boundary layer!), one typically wants x apple L 0 We can express this using the Cell-Peclet number: => Can define Cell-Peclet number: P = x = c x 2L 0 2D = r 2R CFL for advection, e.g. Stability for diffusion, e.g. t r = c x apple 1 R = D t x 2 apple 1 2 The difficulty is now to choose a numerical scheme that is good for advection & diffusion, while keeping P below 1 (to resolve small features, e.g. boundary layers)

6 FD schemes for AD problems u t = cu x + du xx First try: Centered differences: Uj,n+l - Uj,n Uj+l,n - Uj-1,n + = c d A: 4,n At 2Ax AX)^ Stable for R apple 1, P apple1 2 Second order accurate in space Potentially stable for P>1, but oscillations! see Lecture

7 FD schemes for AD problems u t = cu x + du xx Upwind differences (c>0) Stable for 0 apple r 2 apple 2R apple 1 No oscillations for r +2R apple 1, because then all coefficients are positive: First order accuracy Extra diffusion see Lecture

8 Summary Simple explicit schemes: Low accuracy and / or dispersion and / or extra diffusion. Implicit schemes: Possible, unconditionally stable (see e.g. Crank- Nicholson scheme in book), but usually not efficient enough. Another possibility: Do everything explicit, except the problematic diffusion term: Uj,n+l - Uj,n = C Uj+l,n - Uj,n + d a: u,,n+l at Ax (Ax)Z

9 Nonlinear transport & conservation laws What if transport becomes nonlinear?

10 Nonlinear transport & conservation laws A first attempt at understanding what is going on comes from analyzing the characteristic lines Simplest example: lnviscid Burgers' equation We can use the implicit ansatz u(x, t) =u(x ut, 0) du dx Characteristics for initial conditions all meet in (x,t)=(1,1)! u(x, 0) = 1 x Lecture Solution is not defined at (x,t)=(1,1) => so what do we do?

11 Nonlinear transport & conservation laws A more famous example of issues with characteristics is the Riemann problem: Take again Burger s equation: du dx Riemann problem: How does solution evolve in time for initial conditions u(x, 0) = A, x < 0, and u(x, 0) = B, x 0 t t A > B: shock A < B: fan