Summary of explicit methods:AerE 546 Lecture 20

AerE 546 % Lecture 20


Summary of explicit methods:
Euler ! ! ! upwind = central + ε·diffusion, first order accurate
Runge-Kutta !! recall RK is `stable for convectionʼ, time-stepping adds dissipation:
! ! ! variations (low storage, higher damping) are used in CFD.
Lax-Wendroff ! minimum dissipation (ε=C), 2nd order accurate in space and time
MacCormick! ! two-step method, similar to L-W

Generally CFL < 1, or CFLstab → time-step restriction ∆t < min[ CFLstab ∆x / a
NOTE: distinction is stencil, algorithm uses RHS * ∆t


Finite volume thinking: flux interpolation, vs. finite-difference. ∂tu = -∂xF.

A. Finite volume-like semi-discretization
● ╎ ● ╎ ●
∫∂tu dx ≈ ∂tuj ∆x = -[ Fj+1/2 - Fj-1/2 ] j-1 j-½ j j+ ½ j+1
Recall uj is cell average: ∂u/∂t Vj = -Fj+1/2 Aj+1/2 + Fj-1/2 Aj-1/2 . The focus of hyperbolic
numerics is the RHS -- e.g., flux interpolation, limiters, splitting, etc.




F! ? x
! ! |

! x


! | j-½ | |
x j-1 j j+1

What are the cell face fluxes? F= true flux; F̃ = numerical flux
Numerical ≠ True, e.g. upwinding




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, AerE 546 % Lecture 20



B. Estimate of F̃ j-1/2 : Interpolation methods: F = a +b s + c s2 ....




Piecewise constant



! ! ! ! ! ! j-1 j j+1




Piecewise linear



% % % % % %  j j+1/2 j+1




 Fj−1 +O(∆x), u > 0
1st order upwind: F̃j−1/2 = Fj+1 +O(∆x), u < 0






1
2nd order central: F̃j−1/2 = (Fj + Fj−1 ) +O(∆x)2
2





 1

 (3Fj + 6Fj−1 − Fj−2 ), u > 0
8
QUICK: F̃j−1/2 = 1 +O(∆x)3

 (3Fj−1 + 6Fj − Fj+1 ), u < 0

 8





QUICK is quadratic: upwind biased. Lagrange interpolation at j+1/2:
(xj+1 − x)(xj − x) (xj+1 − x)(xj−1 − x) (xj − x)(xj−1 − x)
F = Fj−1 2
− Fj 2
+ Fj+1
2∆x ∆x 2∆x2
Fj−1 3Fj 3Fj+1
Fj+1/2 =− + +
8 4 8

Or fit a quadratic, F = a +b s + c s2
s=0,-1,-2: a= Fj ; Fj-1 = Fj - b+c ; Fj-2 = Fj - 2b+ 4c → Fj-1/2 = Fj - b/2+c/4
Fj-1/2 = Fj (1/2)(3/2)/2 + Fj-1 1/2(3/2) - Fj-2 1/2(1/2)/2 = 3/8Fj + 3/4Fj-1 - 1/8Fj-2




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