Lecture 618.086

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4

Schär, ETH Zürich

t

n+1!

n!

n-1!

n-2!

x i-3! i-2! i-1! i! i+1! i+2! i+3!

CFL criterion for Leapfrog scheme

numerical domain of dependence

physical domain of dependence

!"!t

+ u !"!x

= 0Equation:

!in+1 = !i

n–1 –" !i+1n #!i–1

n( ) with " = u$t$x

Scheme:

stable |uΔt/Δx| ≤ 1

unstable |uΔt/Δx| > 1

Leapfrog scheme• Easiest numerical scheme for 2nd order problem: Leapfrog

Notation:

Uj,n+1 � 2Uj,n + Uj,n�1

�t

2= c

2Uj+1,n � 2Uj,n + Uj�1,n

�x

2

Uj,n = U(j�x, n�t)

• Stability: |r| ≤ 1 (equiv. CFL condition!)

• Accuracy: 2nd order

Lecture

Lecture / see Mathematica notebook leapfrog_stability.nb

c

c

Accuracy of FD schemes: Alternative way (using modified equations)

• Approach is general and gives GLOBAL error directly (not local error)

Uj,n+1 � 2Uj,n + Uj,n�1

�t

2= c

2Uj+1,n � 2Uj,n + Uj�1,n

�x

2

• Using wave equation and leapfrog as an example:

• Plug in analytical u and expand in Taylor on LHS and RHS of FD equation E.g.:uj,n+1 = u(x, t) + u

x

(x, t)�t+1

2utt

(x, t)�t2 +O(�t3)

• Simplifying LHS and RHS givesu

tt

+1

12�t

2u

tttt

+ . . . = c

2(uxx

+1

12�x

2u

xxxx

+ . . .)

• Note that , so the global error isutttt

= c2uttxx

= c4uxxxx

1

12

⇥�t

2c

4 ��x

2⇤u

xxxx

=> p=2 in time and space

Staggered grids• Remember:• Equivalent 1st order problem: @

@t

✓v1

v2

◆=

✓0 c

c 0

◆@

@x

✓v1

v2

◆

with v1 = ut

, v2 = cux

• There are a number of “physical” wave equations that are of this form, most importantly the Maxwell equations of electrodynamics (in vacuum):

Leads to a 1st order system as above!

We could try this….• Discretize 1st order problem:

with 1st order “leapfrog”

This would give us 2nd order accuracy, because we use two centered differences for space and time in leapfrog!

@

@t

✓v

w

◆=

✓0 c

c 0

◆@

@x

✓v

w

◆

Vj,n+1 � Vj,n

2�t

= c

Wj+1,n �Wj�1,n

2�x

Wj,n+1 �Wj,n

2�t

= c

Vj+1,n � Vj�1,n

2�x

• 1way wave eqleapfrog:

Uj,n+1 � Uj,n

2�t

= c

Uj+1,n � Uj�1,n

2�x

• Here:

Lecture: interesting numerical “dispersion”!

• 1-way wave eq. leapfrog: dispersion-free |G|=1 for all k, as long as CFL condition r<=1

correct typo in one-way LF scheme!

1way-Leapfrog• “Coupled” equations decouple in current leapfrog formulation!

Diffusion and Advection (6.5)• The heat equation is another famous PDE

ut

= Duxx

u(x, t) =1p4⇡Dt

e

�x

2

4Dt

• Solution to IV u(x,0) = δ(x) is given by:

• D is the diffusion constant (units length^2/time)

• Growth factor: G(k, t) = e�Dk2t

Numerical scheme (FD)• Heat equation is dissipative, so why not try Forward Euler:

Uj,n+1 � Uj,n

�t

=Uj+1,n � 2Uj,n + Uj�1,n

�x

2

• Expected accuracy: O(Δt) in time, O(Δx2) in space.

• Stability in the usual way gives

• 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!

R =�t

�x

2 1

2

Incorporating boundary conditions into FD matrices

• Writing the heat equation using method of lines (with centered differences in x), we have

~ut = A~u• Away from boundaries, Ai = (0, . . . ,�1, 2,�1, 0, . . . , 0)

position j=i

• Boundary conditions determine values of A near boundaries, e.g.0

BBBBBB@

0 0 0 0 0 0�1 2 �1 0 0 00 �1 2 �1 0 00 0 �1 2 �1 00 0 0 �1 2 �10 0 0 0 0 0

1

CCCCCCA

0

BBBBBB@

u0

u1

u3

. . .uN�1

uN

1

CCCCCCA

BCs:u(0,t)=v1 u(X,t)=v2

see book, 1.2