Presentation Slides for

Chapter 6of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

Stanford UniversityStanford, CA 94305-4020jacobson@stanford.edu

March 10, 2005

Ordinary differential equation (ODE)Equation with one independent variable

ODEs and PDEs

Partial differential equation (PDE)Equation with more than one independent variable

OrderHighest derivative of an equation

DegreeHighest polynomial value of the highest derivative

Initial value problemConditions are known at one end of domain but not other

Boundary value problemConditions are known at both ends of domain

ODEs and PDEs

Operator Splitting Scheme

Fig. 6.1

Dynamics

Transport

Gas chemistry

Time interval 1

Dynamics

Transport

Gas chemistry

Time interval 2

Operator-split advection-diffusion equations (6.1-6.3)

Operator Splitting

Operator-split external source/sink terms (6.4)

∂N∂t

+∂ uN( )

∂x−

∂∂x

Kh,xx∂N∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ =0

∂N∂t

+∂ vN( )

∂y−

∂∂y

Kh,yy∂N∂y

⎛

⎝ ⎜

⎞

⎠ ⎟ =0

∂N∂t

+∂ wN( )

∂z−

∂∂z

Kh,zz∂N∂z

⎛ ⎝ ⎜

⎞ ⎠ ⎟ =0

∂N∂t

= Rnn=1

Ne,t

∑

Replacement of continuous differential operator (d) with discrete difference analog () written in terms of a finite number of values along a temporal or spatial direction.

Examples:

Finite-Difference Approximation

dNdt

→ΔNΔt

∂N∂x

→ΔNΔx

∂u∂x

→ΔuΔx

Finite Difference Example

Fig. 6.2

First, replace continuous function ux with a finite number of values in the x direction.

ux

u

x

i-1 i i+1

ui-1 ui ui+1 ui+2

xi-1 xi xi+1 xi+2

Finite Difference ApproximationSecond, define differences of du at point xi

-->

-->

-->

Δui =ui+1−ui−1

Δui =ui+1−ui

Δui =ui −ui−1 backward difference

forward difference

central difference

Central difference approximation to tangent slope at xi (6.10)

∂u∂x

≈ΔuiΔxi

=ui+1−ui−1xi+1−xi−1

ux

u

x

i-1 i i+1

ui-1 ui ui+1 ui+2

xi-1 xi xi+1 xi+2

Finite-Difference Approximations

Fig. 6.3

Central (AC), forward (BC), and backward (AB) approximations to slope of tangent at point B

Nx

N

x

xi-1 xi xi+1

A

B

C

x- x x x+x

x x

Taylor series expansion at point x+x (6.11)

Taylor Series Expansion

Taylor series expansion at point x-x (6.12)

Nx+Δx =Nx +Δx∂Nx∂x

+12

Δx2∂2Nx∂x2 +

16

Δx3∂3Nx∂x3

+124

Δx4∂4Nx∂x4

+...

Nx−Δx =Nx −Δx∂Nx∂x

+12

Δx2∂2Nx∂x2

−16

Δx3∂3Nx∂x3

+124

Δx4∂4Nx∂x4

−...

Nx

N

x

xi-1 xi xi+1

A

B

C

x- x x x+x

x x

Sum the Taylor series expansions (6.13)

Finite Difference Approximations

Rearrange (6.14)

Truncation error (neglect 2nd-order terms and higher) (6.15)

2nd-order central difference approx. of 2nd derivative (6.16)

Nx+Δx +Nx−Δx =2Nx+Δx2 ∂2Nx∂x2 +

112

Δx4∂4Nx∂x4 +...

∂2Nx∂x2

=Nx+Δx−2Nx+Nx−Δx

Δx2+O Δx2( )

O Δx2( ) =−112

Δx2∂4Nx∂x4

−...

∂2Nx∂x2

≈Nx+Δx−2Nx+Nx−Δx

Δx2

Subtract the Taylor series expansions (6.17)

Finite Difference Approximations

Rearrange (6.18)

Truncation error (6.19)

2nd-order central difference approx. of 1st derivative (6.20)

Nx+Δx −Nx−Δx =2Δx∂Nx∂x

+13

Δx3∂3Nx∂x3

+...

∂Nx∂x

=Nx+Δx−Nx−Δx

2Δx+O Δx2( )

O Δx2( ) =−16

Δx2∂3Nx∂x3

−...

∂Nx∂x

≈Nx+Δx−Nx−Δx

2Δx=Ni+1−Ni−1

2Δx

Finite Difference Approximations

1st-order backward difference approx. of 1st derivative (6.22)

First two terms of Taylor series

1st-order forward difference approx. of 1st derivative (6.21)

∂Nx∂x

≈Nx+Δx−Nx

Δx=Ni+1−Ni

Δx

∂Nx∂x

≈Nx −Nx−Δx

Δx=Ni −Ni−1

Δx

Differencing Time Derivative

Central, forward, backward difference approximations (6.23)

∂Nt∂t

≈Nt+h −Nt−h

2h

∂Nt∂t

≈Nt+h −Nt

h

∂Nt∂t

≈Nt −Nt−h

h

Convergence of finite difference analog (6.6)

Consistency, Convergence

Consistency of finite difference analog (6.7)

Convergence of overall solution (6.8)

∂N∂x

= limΔx→ 0

ΔNΔx

limΔx→ 0

T.E.ΔNΔx

⎛ ⎝ ⎜ ⎞

⎠ ⎟ =0

limΔx,Δt→ 0

Ne,x,t −Nf,x,t =0

StabilityStability (6.9)lim

t→ ∞Ne,x,t −Nf,x,t ≤C

Conditionally stable: Stable for limited time-step rangeUnconditionally stable: Stable for all time stepsUnconditionally unstable: Unstable for all time steps

An unconditionally unstable scheme cannot be convergent overall, but individual finite-difference analogs in an unstable scheme may converge and may be consistent.

In other words, consistency and convergence of individual analogs do not guarantee stability.

Stability is guaranteed if a scheme is convergent overall and its finite-difference analogs are convergent and consistent.

Numerical Diffusion, DispersionNumerically diffusive scheme:

Peaks spread artificially across grid cells

Numerically dispersive (oscillatory) scheme: Waves appear ahead of or behind peaks

Unbounded scheme: Transported values artificially rise above the largest existing value or fall below the smallest existing value in domain.

Nonmonotonic scheme: Gradients are not preserved during transport

High Order Approximations

Finite difference approximation of ∂mN/∂xm

• Order of derivative = m• Approximation expanded across q discrete nodes• Minimum number of nodes = m + 1• Maximum order of approximation = q - m

ExampleOrder of derivative: m = 1Number of nodes: q = 5

--> Order of approximation: q - m = 4

High Order Approximations

Grid spacing where q=5. Derivative is taken at x3. Fig. 6.4

Distance between two nodes

Approximation to the mth derivative across q nodes (6.24)

x

x1 x2 x3 x4 x5

*

Δxi =xi+1−xi

∂mN

∂xm≈ γiNii=1

q

∑ =γ1N1+γ2N2 +...+γqNq

High Order Approximations

Taylor series expansion for each node about point x* (6.25)

Combine (6.24) with (6.25) and gather terms (6.26)

Redefine (6.27)

Ni =N* + xi −x*( )∂N*∂x

+12xi −x*( )

2 ∂2N*∂x2 +

16xi −x*( )

3 ∂3N*∂x3

+...

∂mN

∂xm≈ γiNii=1

q

∑ = γiN*i=1

q

∑ + γi xi −x*( )∂N*∂x

i=1

q

∑ + γi12xi −x*( )

2 ∂2N*∂x2

i=1

q

∑ +...

γiNii=1

q

∑ =B0N* +B1∂N*∂x

+B2∂2N*∂x2

+...

Approximation to the mth derivative across q nodes (6.24)

∂mN

∂xm≈ γiNii=1

q

∑

Bn = γi1n!

xi −x*( )n

i=1

q

∑

High Order Approximations

for n = 0…q - 1 (6.28)

• Set Bn=0 for all n, except n = m• Set Bn=1 when n = m

Multiply (6.28) through by n! and set matrix (6.29)

Bn = γi1n!

xi −x*( )n

i=1

q

∑

1 1 1 ... 1

x1−x*( ) x2 −x*( ) x3−x*( ) ... xq −x*( )

x1−x*( )2 x2 −x*( )

2 x3−x*( )2 ... xq −x*( )

2

: : : :

x1−x*( )q−1 x2−x*( )

q−1 x3 −x*( )q−1 ... xq −x*( )

q−1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

γ1γ2γ3:

γq

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

0!B01!B12!B2

:

q−1( )!Bq−1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

2nd-Order Central Diff. Approx.

Find second-order central difference approx. to ∂N/∂x

Order of derivative: m = 1Order of approximation: q - m = 2

--> Number of nodes: q = 3

Example

Set matrix(6.32)

1 1 1

−Δx 0 Δx

−Δx( )2 0 Δx( )2

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

γi−1γi

γi+1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

=

0

1

0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

2nd-Order Central Diff. Approx.

Apply the 's to (6.24)

Substitute 's to obtain central difference approx. Table 6.2 (c)

∂N∂x

≈γ1N1+γ2N2 +γ3N3 =γi−1Ni−1+γiNi +γi+1Ni+1

∂N∂x

≈Ni+1−Ni−1

2Δx

γi+1 =1

2Δxγi−1=−

12Δx

γi =0

Solve matrix

1st-Order Backward Diff. Approx.

Find first-order backward difference approx. to ∂N/∂x

Order of derivative: m = 1Order of approximation: q - m = 1

--> Number of nodes: q = 2

Example

Set matrix(6.30)

1 1

−Δx 0⎡

⎣ ⎢ ⎤

⎦ ⎥ γi−1γi

⎡

⎣ ⎢ ⎤

⎦ ⎥ =0

1⎡

⎣ ⎢ ⎤

⎦ ⎥

1st-Order Backward Diff. Approx.

Apply the 's to (6.24)

Substitute 's to obtain backward difference approx. Table 6.2 (a)

Solve matrix

γi−1=−1Δx

γi =1Δx

∂N∂x

≈γi−1Ni−1+γiNi

∂N∂x

≈Ni −Ni−1

Δx

2nd-Order Backward Diff. Approx.

Find second-order backward difference approx. to ∂N/∂x

Order of derivative: m = 1Order of approximation: q - m = 2

--> Number of nodes: q = 3

Example

Set matrix (6.32)1 1 1

−2Δx −Δx 0

−2Δx( )2 −Δx( )2 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

γi−2γi−1γi

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ =

0

1

0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

∂N∂x

≈Ni−2 −4Ni−1+3Ni

2Δx

Solve (Table 6.2d)

Higher-Order Approximations

Third-order forward difference (m = 1, q = 4) Table 6.2 (g)

Fourth-order backward difference (m = 1, q = 5) Table 6.2 (i)

Third-order backward difference (m = 1, q = 4) Table 6.2 (f)

Fourth-order forward difference (m = 1, q = 5) Table 6.2 (j)

∂N∂x

≈Ni−2 −6Ni−1+3Ni +2Ni+1

6Δx

∂N∂x

≈−2Ni−1−3Ni +6Ni+1−Ni+2

6Δx

∂N∂x

≈−Ni−3+6Ni−2 −18Ni−1+10Ni +3Ni+1

12Δx

∂N∂x

≈−3Ni−1−10Ni +18Ni+1−6Ni+2 +Ni+3

12Δx

Fourth-Order Approximations

Fourth-order backward diff. scheme (m = 1, q = 5) Table 6.2 (k)

Discretize around furthest cell

Fourth-order forward difference (m = 1, q = 5) Table 6.2 (l)

1 1 1 1 1

−4Δx −3Δx −2Δx −Δx 0

−4Δx( )2 −3Δx( )2 −2Δx( )2 −Δx( )2 0

−4Δx( )3 −3Δx( )3 −2Δx( )3 −Δx( )3 0

−4Δx( )4 −3Δx( )4 −2Δx( )4 −Δx( )4 0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

γi−4γi−3γi−2γi−1γi

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

0

1

0

0

0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

∂N∂x

≈−3Ni−4+16Ni−3 −36Ni−2 +48Ni−1−25Ni

12Δx

∂N∂x

≈25Ni −48Ni+1+36Ni+2 −16Ni+3+3Ni+4

12Δx

Fourth-Order Central Diff. Approx.

Fourth-order central difference of ∂N/∂x (m = 1, q = 5) (6.33)

Table 6.2 (h)

1 1 1 1 1

−2Δx −Δx 0 Δx 2Δx

−2Δx( )2 −Δx( )2 0 Δx( )2 2Δx( )2

−2Δx( )3 −Δx( )3 0 Δx( )3 2Δx( )3

−2Δx( )4 −Δx( )4 0 Δx( )4 2Δx( )4

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

γi−2γi−1γi

γi+1γi+2

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

0

1

0

0

0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

∂N∂x

≈Ni−2 −8Ni−1+8Ni+1−Ni+2

12Δx

Fourth-Order Central Diff. Approx.

Fourth-order central difference of ∂2N/∂x2 (m = 2, q = 5)

Table 6.2 (n)

1 1 1 1 1

−2Δx −Δx 0 Δx 2Δx

−2Δx( )2 −Δx( )2 0 Δx( )2 2Δx( )2

−2Δx( )3 −Δx( )3 0 Δx( )3 2Δx( )3

−2Δx( )4 −Δx( )4 0 Δx( )4 2Δx( )4

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

γi−2γi−1γi

γi+1γi+2

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

0

0

2

0

0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

∂2N

∂x2 ≈−Ni−2 +16Ni−1−30Ni +16Ni+1−Ni+2

12Δx2

Advection-Diffusion Eqn. SolutionsSpecies continuity equation in west-east direction (6.1)

CFL stability criterion for advection

Stability criterion for diffusion

∂N∂t

+∂ uN( )

∂x−

∂∂x

Kh,xx∂N∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ =0

h<Δxmin umax

h<Δxmin2 Kmax

Example:xmin=5 km, |umax|=20 m/s --> h=250 s (Hydrostatic model)xmin=5 km, |umax|=346 m/s --> h=14.5 s (Nonhydrostatic model)

Example:zmin=20 m, Kmax=50 m2/s --> h=8 s (in the vertical)

FTCS SolutionForward in time, centered in space (FTCS) solution (6.35)

1st-order approximation in time

Unconditionally unstable for K=0, large K; otherwise conditionally stable

Ni,t −Ni,t−hh

+uN( )i+1,t−h − uN( )i−1,t−h

2Δx

−KNi+1,t−h −2Ni,t−h +Ni−1,t−h

Δx2 =0

Advection-Diffusion Eqn. SolutionsImplicit solution: 1st-order approximation in time (6.36)

Unconditionally stable for all u, K

Rearrange and write in tridiagonal matrix form (6.37)

Ai =−hu

2Δx+

K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i−1

Bi =1+h2K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i

Di =hu

2Δx−

K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i+1

(6.38)

Ni,t −Ni,t−hh

+uN( )i+1,t − uN( )i−1,t

2Δx−K

Ni+1,t −2Ni,t +Ni−1,t

Δx2=0

AiNi−1,t +BiNi,t +DiNi+1,t =Ni,t−h

Advection-Diffusion Eqn. SolutionsRearrange and write in tridiagonal matrix form (6.39)

B1 D1 0 0 ... 0 0 0

A2 B2 D2 0 ... 0 0 0

0 A3 B3 D3 ... 0 0 0

0 0 A4 B4 ... 0 0 0

: : : : : : :

0 0 0 0 ... BI−2 DI−2 0

0 0 0 0 ... AI−1 BI−1 DI−10 0 0 0 ... 0 AI BI

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

N1,tN2,tN3,tN4,t

:

NI −2,tNI−1,tNI,t

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

N1,t−hN2,t−hN3,t−hN4,t−h

:

NI−2,t−hNI−1,t−hNI ,t−h

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

−

A1N0,t0

0

0

:

0

0

DINI+1,t

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Tridiagonal Matrix SolutionMatrix decomposition: (6.40)

i = 2..I

i = 2..I

Matrix backsubstitution: (6.41)

i = I -1..1, -1

γ1 =−D1B1

γi =−Di

Bi +Aiγi−1

α1 =R1B1

αi =Ri −Aiαi−1Bi +Aiγi−1

NI ,t =α I Ni,t =αi +γiNi+1,t

Tridiagonal Matrix SolutionMatrix for solution for periodic boundary conditions (6.42)

B1 D1 0 0 ... 0 0 A1A2 B2 D2 0 ... 0 0 0

0 A3 B3 D3 ... 0 0 0

0 0 A4 B4 ... 0 0 0

: : : : : : :

0 0 0 0 ... BI−2 DI−2 0

0 0 0 0 ... AI−1 BI −1 DI −1DI 0 0 0 ... 0 AI BI

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

N1,tN2,tN3,tN4,t

:

NI−2,tNI −1,tNI ,t

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

N1,t−hN2,t−hN3,t−hN4,t−h

:

NI−2,t−hNI−1,t−hNI,t−h

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Values at node I are adjacent to those at node 1

Crank-Nicolson SchemeCrank-Nicolson form (6.44)

c = Crank-Nicolson parameter= 0.5 --> Crank-Nicolson solution (unconditionally stable all u, K)

2nd-order approximation in time = 0. --> explicit (FTCS) solution= 1 --> implicit solution

Ni,t −Ni,t−hh

+ μcuN( )i+1,t − uN( )i−1,t

2Δx+ 1−μc( )

uN( )i+1,t−h− uN( )i−1,t−h2Δx

⎡

⎣ ⎢

⎤

⎦ ⎥

−K μcNi+1,t −2Ni,t +Ni−1,t

Δx2+ 1−μc( )

Ni+1,t−h −2Ni,t−h +Ni−1,t−h

Δx2⎡

⎣ ⎢ ⎤

⎦ ⎥ =0

Crank-Nicolson SchemeTridiagonal form (6.45)

Ai =−μchu

2Δx+

K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i−1

Ei = 1−μc( )hu

2Δx+

K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i−1

Bi =1+μch2K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i

Fi =1− 1−μc( )h2K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i

Di =μchu

2Δx−

K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i+1

Gi =−1−μc( )hu

2Δx−

K

Δx2⎛

⎝ ⎜

⎞

⎠ ⎟ i+1

AiNi−1,t +BiNi,t +DiNi+1,t =EiNi−1,t−h +FiNi,t−h+GiNi+1,t−h

Leapfrog Scheme2nd-order approximation in time (6.48)

Unconditionally unstable for all nonzero K; conditionally stable for linear equations when K=0; unstable for nonlinear equations

Ni,t −Ni,t−2h2h

+uN( )i+1,t−h − uN( )i−1,t−h

2Δx−K

Ni+1,t−h −2Ni,t−h +Ni−1,t−h

Δx2=0

Matsuno Scheme

Prediction step (6.49)

Correction step (6.49)

Ni,est−Ni,t−hh

+uN( )i+1,t−h −uN( )i−1,t−h

2Δx−K

Ni+1,t−h−2Ni,t−h+Ni−1,t−h

Δx2=0

Ni,t −Ni,t−hh

+uN( )i+1,est− uN( )i−1,est

2Δx−K

Ni+1,est−2Ni,est+Ni−1,est

Δx2=0

1st-order approximation in time

Conditionally stable for all u when K=0 or small; unconditionally unstable for large K

Heun Scheme2nd-order approximation in time (6.51)

Unconditionally unstable for all u when K=0 and K large; conditionally stable for other values of K

Ni,t −Ni,t−hh

+12

uN( )i+1,est− uN( )i−1,est2Δx

−K2

Ni+1,est−2Ni,est+Ni−1,est

Δx2

+12

uN( )i+1,t−h − uN( )i−1,t−h2Δx

−K2

Ni+1,t−h −2Ni,t−h +Ni−1,t−h

Δx2 =0

Prediction step same as first Matsuno step

Adams-Bashforth Scheme2nd-order approximation in time (6.52)

Unconditionally unstable for all u when K=0 and K large; conditionally stable for other values of K.

Ni,t −Ni,t−hh

+32

uN( )i+1,t−h −uN( )i−1,t−h2Δx

−32KNi+1,t−h −2Ni,t−h +Ni−1,t−h

Δx2

−12

uN( )i+1,t−2h − uN( )i−1,t−2h2Δx

+12KNi+1,t−2h −2Ni,t−2h +Ni−1,t−2h

Δx2=0

Fourth-Order Runge-Kutta SchemeConditionally stable (6.53-5)

Ni,t =Ni,t−h+k16

+k23

+k33

+k46

k1 =h −uN( )i+1,t−h− uN( )i−1,t−h

2Δx+K

Ni+1,t−h−2Ni,t−h+Ni−1,t−h

Δx2⎡

⎣ ⎢

⎤

⎦ ⎥

k2 =h −ut−hNest1( )i+1−ut−hNest1( )i−1

2Δx+K

Ni+1,est1−2Ni,est1+Ni−1,est1

Δx2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Ni,est1=Ni,t−h +k12

Ni , est 2

= Ni , t − h

+

k2

2

Fourth-Order Runge-Kutta Scheme

Ni,est3 =Ni,t−h +k3

k4 =h −ut−hNest3( )i+1−ut−hNest3( )i−1

2Δx+K

Ni+1,est3−2Ni,est3+Ni−1,est3

Δx2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

k3=h −ut−hNest2( )i+1−ut−hNest2( )i−1

2Δx+K

Ni+1,est2 −2Ni,est2+Ni−1,est2

Δx2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Convergence of Four Schemes

(Ketefian and Jacobson 2004b) Fig. 6.5

10

-11

10

-9

10

-7

10

-5

10

-3

10

-1

0.1 1 10

Runge-Kutta

Adams-Bashforth

Matsuno

Forward Euler

Error

Time step (s)

1

st

-order

5

th

-6

th

order

2

nd

-order

Err

or

Fourth-Order in Space SolutionCentral difference implicit solution (6.56)

Write in Crank-Nicolson and pentadiagonal form

Ni,t −Ni,t−hh

+uN( )i−2,t −8uN( )i−1,t +8uN( )i+1,t − uN( )i+2,t

12Δx

−K−Ni−2,t +16Ni−1,t −30Ni,t +16Ni+1,t −Ni+2,t

12Δx2 =0

AiNi−2,t+BiNi−1,t +DiNi,t +EiNi+1,t +FiNi+2,t

=PiNi−2,t−h+QiNi−1,t−h +SiNi,t−h+TiNi+1,t−h +UiNi+2,t−h

Variable Grid Spacing, WindsSecond-order central difference of advection term (6.57)

Solve matrix equation to obtain coefficients (6.58)

∂ uN( )∂x

=γa,i−1 uN( )i−1+γa,i uN( )i +γa,i+1 uN( )i+1

1 1 1

−xi −xi−1( ) 0 xi+1−xi( )

xi −xi−1( )2 0 xi+1−xi( )

2

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

γa,i−1γa,i

γa,i+1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

=

0

1

0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Variable Grid Spacing, Winds

Solve matrix equation to obtain coefficients (6.59)

γa,i+1 =xi −xi−1

xi+1−xi( ) xi+1−xi−1( )

γa,i =xi+1−xi( ) − xi −xi−1( )xi+1−xi( ) xi −xi−1( )

γa,i−1 =−xi+1−xi( )

xi −xi−1( ) xi+1−xi−1( )

Variable Grid Spacing, DiffusionExpand analytical diffusion term (6.60)

Second-order central-difference approx. to terms (6.61)

∂∂x

K∂N∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ =

∂K∂x

∂N∂x

+K∂2N

∂x2

∂K∂x

≈γa,i−1Ki−1+γa,iKi +γa,i+1Ki+1

∂N∂x

≈γa,i−1Ni−1+γa,iNi +γa,i+1Ni+1

K∂2N

∂x2≈Ki γd,i−1Ni−1+γd,iNi +γd,i+1Ni+1( )

Variable Grid Spacing, DiffusionSolve matrix equation to obtain coefficients (6.63)

γd,i−1=2

xi −xi−1( ) xi+1−xi−1( )

γd,i =−2

xi+1−xi( ) xi −xi−1( )

γd,i+1=2

xi+1−xi( ) xi+1−xi−1( )

Variable Grid Spacing SolutionCrank-Nicolson form (c=0.5--> 2nd order time, space) (6.66)

Coefficients for diffusion term(6.65)

Write in tridiagonal form

Ni,t −Ni,t−hh

=−μc γau−βK( )N[ ]i−1+ γau−βK( )N[ ]i + γau−βK( )N[ ]i+1{ }t

−1−μc( ) γau−βK( )N[ ]i−1+ γau−βK( )N[ ]i + γau−βK( )N[ ]i+1{ }t−h

βK ,i−1 = γa,i−1Ki−1+γa,iKi +γa,i+1Ki+1( )γa,i−1+Kiγd,i−1

βK ,i = γa,i−1Ki−1+γa,iKi +γa,i+1Ki+1( )γa,i +Kiγd,i

βK ,i+1 = γa,i−1Ki−1+γa,iKi +γa,i+1Ki+1( )γa,i+1+Kiγd,i+1

AiNi−1,t +BiNi,t +DiNi+1,t =EiNi−1,t−h +FiNi,t−h+GiNi+1,t−h

Two Dimensional Solution Advection-diffusion equation in two dimensions

(6.67)

Central difference approximation(6.68)

Solve

∂N∂t

+∂ uN( )

∂x+

∂ vN( )∂y

−∂∂x

Kh,xx∂N∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ −

∂∂y

Kh,yy∂N∂y

⎛

⎝ ⎜

⎞

⎠ ⎟ =0

Ni, j,t −Ni,j,t−hh

+uN( )i+1, j − uN( )i−1, j

2Δx+vN( )i, j+1− vN( )i,j−1

2Δy

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ t

− Kh,xxNi−1, j −2Ni, j +Ni+1, j

Δx2 +Kh,yyNi, j−1−2Ni, j +Ni, j+1

Δy2⎛

⎝ ⎜

⎞

⎠ ⎟ t=0

Ai, jNi−1, j,t +Bi, jNi, j,t +Di, jNi+1,j,t +Ei, jNi, j−1,t +Fi,jNi, j+1,t =Ni, j,t−h

Semi-Lagrangian Method

Nx,t =Nx−uh,t−h

Concentration at current time t and existing node x (6.69)

Values at time x-uh, t-h interpolated from existing nodes

Finite Element Method Advection equation at node i

(6.70)

Trial function = series expansion approximation to N (6.71) = linear combination of basis functions

ej(x) = basis function

j = trial space

∂Ni∂t

+∂ uN( )i

∂x=0

Ni ≈Ni x( ) = Njej x( )

j∑

Finite Element Method

Minimize residual (6.72) Force its weighted average to zero over domain

Residual in advection equation (6.73)

ei(x) = weight function

ei(x) = ej(x) --> Galerkin method of weighted residuals

ei(x) ≠ ej(x) --> Petrov-Galerkin technique

Ri x( )ei x( )dxx∫ =0

Ri x( ) =∂Ni x( )

∂t+u

∂Ni x( )∂x

⎡

⎣ ⎢ ⎤

⎦ ⎥ −∂Ni∂t

+u∂Ni∂x

⎡ ⎣ ⎢

⎤ ⎦ ⎥ =

∂Ni x( )∂t

+u∂Ni x( )

∂x⎡

⎣ ⎢ ⎤

⎦ ⎥ −0

Finite Element Method Substitute

∂Ni x( )∂t

+u∂Ni x( )

∂x⎡

⎣ ⎢ ⎤

⎦ ⎥ ei x( )dxx∫

=∂∂t

Njej x( )

j∑

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟ +u

∂∂x

Njej x( )

j∑

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ei x( )dx

x∫

=∂Nj∂t

ej x( )ei x( )dxx∫

⎛

⎝ ⎜

⎞

⎠ ⎟

j∑ +u Nj

dej x( )

dxei x( )dx

x∫⎛

⎝ ⎜

⎞

⎠ ⎟

j∑ =0

Ri x( )ei x( )dxx∫ =0

Ri x( ) =∂Ni x( )

∂t+u

∂Ni x( )∂x

Ni x( )= Njej x( )

j∑

into

to obtain (6.74)

Finite Element Method Take time difference of (6.74) over three nodes

(6.75)Ni−1,t −Ni−1,t−h

hei−1 x( )ei x( )xi−1

xi∫ dx+Ni,t −Ni,t−h

hei x( )ei x( )xi−1

xi+1∫ dx

+Ni+1,t −Ni+1,t−h

hei+1 x( )ei x( )xi

xi+1∫ dx

+u

Ni−1,tdei−1 x( )

dxei

xi−1

xi∫ x( )dx+Ni,t

dei x( )dx

ei x( )xi−1

xi+1

∫ dx

+Ni+1,tdei+1 x( )

dxei

xi

xi+1

∫ x( )dx

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

=0

Finite Element Method Define basis functions as chapeau functions

(6.76)

Solve each integral(6.77)

ei x( )=

x−xi−1xi −xi−1

xi−1≤x≤xi

xi+1−xxi+1−xi

xi ≤x≤xi+1

0 all other cases

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

ei−1 x( )ei x( )xi−1

xi∫ dx=

xi −xxi −xi−1

⎛

⎝ ⎜

⎞

⎠ ⎟

x−xi−1xi −xi−1

⎛

⎝ ⎜

⎞

⎠ ⎟

xi−1

xi∫ dx=

x−xi−16

Finite Element Method Substitute integral solutions back into (6.75)

(6.78)Ni−1,t −Ni−1,t−h( )Δxi + Ni,t −Ni,t−h( )2 Δxi+1+Δxi( ) + Ni+1,t −Ni+1,t−h( )Δxi+1

6h

+uNi+1,t −Ni−1,t

2=0

Solve with tridiagonal matrix solution --> 4th-order in space, 2nd order in time

Tests with a Finite Element Method Preservation of a Gaussian peak during finite element transport after

eight revolutions around a circular grid when uh/x= 0.02

-200

0

200

400

600

800

1000

1200

0 5 10 15 20 25 30

Concentration (generic)

Grid cell number

Fig. 6.8

Con

cent

ratio

n

Tests with a Finite Element Method Preservation of a Gaussian peak during finite element transport after

eight revolutions around a circular grid when uh/x= 0.6

-400

-200

0

200

400

600

800

1000

1200

0 5 10 15 20 25 30

Concentration (generic)

Grid cell number Fig. 6.8

Con

cent

ratio

n

Pseudospectral Method Advection equation

(6.87)∂N∂t

+u∂N∂x

=0

Represent solution with infinite Fourier series(6.88)

N x,t( )= ak t( )eik2πx L

k=0

∞

∑For t=0, integrate both sides of (6.88) from 0≤x≤L

(6.89)

ak 0( )=1L

N x,0( )0

L

∫ e−ik2πx Ldx

Pseudospectral Method Truncate infinite series

(6.90)

Central time-difference approximation of (6.90)(6.91)

Partial derivative of (6.90) with respect to space(6.92)

N x,t( )= ak t( )eik2πx L

k=0

K

∑

∂N∂t

≈12h

ak,teik2πx L

k=0

K

∑ − ak,t−2heik2πx L

k=0

K

∑⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

∂N∂x

=ik2πak,t−h

Leik2πx L

k=0

K

∑

Pseudospectral Method

Substitute (6.91) and (6.92) into (6.87)(6.93)

Separate into K equations --> solve(6.94)

12h

ak,t−ak,t−2h( )eik2πx L

k=0

K

∑ =−uik2πak,t−h

Leik2πx L

k=0

K

∑

ak,t −ak,t−2h2h

=−uik2πak,t−h

L

Traits of a Good Advection Scheme

Final shape after six rotations around a 2-D grid. Walcek (2000)

Fig 6.9

BoundedNonoscillatoryMonotonic