ϕ( r ) n ds= 4 πq ijk h x (x i+1/2, y j, z k ) ϕ y (x i, y j+1/ 2, z k ) ϕ z ( x i, y j, z k+1 /2 ) ϕ

Poisson equation

∇

²

ϕ(⃗r)=−4 πρ(⃗r)

^(CGS)

Finite volume method

General approach: divide the space into small volumes and look at the fluxes of some quantity between the volumes.

In electrostatics, we have Gauss theorem:

∮

_∂V

E(⃗r)⋅⃗n dS =− ^⃗ ∮

_∂V

∇ ϕ(⃗r )⋅⃗n dS =4 π ∫

_V

^ρ(⃗r)dV

Consider division into small cubic volumes V_ijk.

∮

_∂V

∇ ϕ(⃗r)⋅⃗n dS =−4 πQ

_ijk

i j

i-1 i+1

j-1

j+1

∮

_∂V

^{∇ ϕ⋅⃗n dS≈}

h

²

( ^{∂ ϕ ∂ x ( x}

^{i+1/ 2}

^{, y}

^j

^{, z}

^k

^{)− ∂ ∂ ϕ x ( x}

^{i−1/ 2}

^{, y}

^j

^{, z}

^k

⁾

+ ∂ ϕ

∂ y ( x

_i

, y

_{j +1/ 2}

, z

_k

)− ∂ ϕ

∂ y ( x

_i

, y

_{j−1 /2}

, z

_k

) + ∂ ϕ

∂ z ( x

_i

, y

_j

, z

_{k+1 /2}

)− ∂ ϕ

∂ z ( x

_i

, y

_j

, z

_{k −1/ 2}

) )

1

∮_∂V

∇ ϕ⋅⃗n dS≈−h

²

( ^{∂ ϕ ∂ x ( x}

^{i+1/ 2}

^{, y}

^j

^{, z}

^k

^{)− ∂ ϕ ∂ x ( x}

^{i−1/ 2}

^{, y}

^j

^{, z}

^k

^{)+ ∂ ∂ ϕ y ( x}

ⁱ

^{, y}

^{j+1/ 2}

^{, z}

^k

⁾

− ∂ ϕ

∂ y ( x

_i

, y

_j−1/2

, z

_k

)+ ∂ ϕ

∂ z ( x

_i

, y

_j

, z

_{k+1/ 2}

)− ∂ ϕ

∂ z ( x

_i

, y

_j

, z

_{k−1/ 2}

) )

Approximate by centred differences:

∮

_∂V ^{∇ ϕ⋅⃗n dS≈ h}

[

^ϕ(xi +1, y_j , z_k)−ϕ(x_i , y _j , z_k)−ϕ(x_i, y _j , z_k)

+ϕ( x

_i−1

, y

_j

, z

_k

)+ϕ( x

_i

, y

_j+1

, z

_k

)−ϕ( x

_i

, y

_j

, z

_k

)

−ϕ( x

_i

, y

_j

, z

_k

)+ϕ( x

_i

, y

_{j −1}

, z

_k

)+ϕ( x

_i

, y

_j

, z

_{k +1}

)

−ϕ( x

_i

, y

_j

, z

_k

)−ϕ( x

_i

, y

_j

, z

_k

)+ϕ( x

_i

, y

_j

, z

_k−1

) ] ^{≈−4 π Q}

ijk

Q

_ijk

≈ h

³

ρ( x

_i

, y

_j

, z

_k

)

gives the standard discretization of the Poisson eq.

So what are the advantages?

1) the volumes don't need to be cubic – can have some arbitrary mesh, like in finite element methods;

2) easy treatment of discrete charges, surface charges, etc.

3) variable ε, including discontinuities:

∮

_∂V

ϵ(⃗r)∇ ϕ(⃗r)⋅⃗n dS =−4 π Q

_ijk

2

Initial-value problems for PDEs

Parabolic equations Diffusion equation:

∂ c(⃗r , t)

∂ t =∇⋅[ D(⃗r)∇ c(⃗r , t)]

Can be obtained from Fick's first law: flux

⃗ J (⃗r , t)=−D(⃗r)∇ c(⃗r , t).

c is the particle concentration

Total particle number

N (t)=

^∫

c(⃗r , t)dV dN (t)

dt = ∂

^∫

c(⃗r , t)dV

∂ t =

^∫

∇⋅[ D(⃗r) ∇ c(⃗r , t)]dV =

^∮_∂V

D(⃗r) ⃗J (⃗r ,t)⋅⃗n dS

=

^∮_∂V

∂ c(⃗r , t)

∂ n dS.

Boundary conditions for space variables:

c(⃗r , t) ∣

_Γ

= f (⃗r , t)

(Dirichlet) or

∂ c(⃗r ,t )

∂ n ∣

Γ

= f (⃗r ,t )

(Neumann)

For Neumann BC with f=0 N is conserved (reflecting BC). Dirichlet BC are also called absorbing.

Initial condition for the time variable:

c(⃗r ,0)=c

₀

(⃗r).

3

∂ u(⃗r , t)

∂ t =∇⋅[ D(⃗r)∇ u(⃗r , t)]

Separation of variables

u (⃗r , t)=R(⃗r)T (t).

R(⃗r) dT (t)

dt =T (t)∇⋅[ D(⃗r)∇ R(⃗r)] ⇒ T ' (t)

T (t) = ∇⋅[ D(⃗r)∇ R(⃗r)]

R(⃗r)

T ' (t)=−λ T (t) ⇒ T (t)=T

₀

e

^−λt

. ∇⋅[ D(⃗r)∇ R(⃗r)]=−λ R(⃗r).

u(⃗r , t) ∣

_Γ

=0 or ∂ u (⃗r ,t)

∂ n ∣

Γ

=0

Elliptic eigenvalue problem with the same boundary conditions. Suppose we know the solution. If the eigenvalues are λ_n and the corresponding

eigenfunctions are then the general solution is

R

_n

(⃗r), u (⃗r , t)=

^∑_n

C

_n

R

_n

(⃗r)e

^−λnt

.

The mode with the smallest λ_n ≡ λ₁ decays the slowest. Conversely, if we solve the parabolic equation corresponding to the given elliptic equation

starting with an arbitrary initial condition, then, unless the initial condition has no “overlap” with R₁, at long times

u (⃗r , t)∼R

₁

(⃗r) e

^−λ1t

.

∫

R

_m

(⃗r) R

_n

(⃗r) dV =0 , λ

_m

≠λ

_n 4

Numerical methods

∂ u(⃗r , t)

∂ t =∇⋅[ D(⃗r)∇ u(⃗r , t)]

Discretize in space, but not in time. Semi-discrete problem.

For simplicity, 1D, D = const.

∂ u( x , t)

∂ t = D ∂

²

u (x , t)

∂ x

²

d u

_m

(t)

d t = D u

_m+1

(t )−2 u

_m

( t)+u

_m−1

( t)

h

²

^u

⁰

^{( t)=0 , u}

^M

^(t)=0

A set of M-1 ODEs. In principle, can use standard methods for solving ODEs.

General principles: 1) different methods have different orders of accuracy; 2) explicit methods have a limited range of stability; 3) some implicit methods are always stable.

Particularly important, because the resulting system turns out to be stiff.

Method of lines.

5

∂ u( x , t)

∂ t = D ∂

²

u (x , t)

∂ x

²

u (x ,t)=u(x)e

^−λt

D d

²

u (x)

d x

²

=−λ u (x) u (x)=sin kx λ= Dk

²

For u(0,t)=u(L , t)=0 , u

_n

( x)=sin π n

L x d u

_m

(t)

d t = D u

_m+1

(t )−2 u

_m

( t)+u

_m−1

( t) h

²

Semi-discrete equation

Since this is a linear system with constant coefficients, there should also be solutions of the form

e

^−λt

.

Guess:

u

_m

(t)=e

^{ikhm−λ t}

. −λ e

^{ikhm−λ t}

= D

h

²

(e

ikh(m+1)−λ t

− 2 e

^{ikhm−λ t}

+e

ikh(m−1)−λ t

)

−λ= D

h

²

(e

^ikh

− 2+e

^−ikh

) ⇒ λ= 2 D

h

²

(1−cos kh)= 4 D

h

²

sin

²

kh 2

λ≈ 2 D

h

²

(( kh)

²

/ 2−(kh)

⁴

/ 24+...)=Dk

²

(1+O ((kh)

²

)) .

All modes decay.

Largest k is π/h; then repeats periodically. λ = 4D/h². Smallest k is π/h. λ = π²D/L². Ratio of the largest and smallest λ is ~L²/h². Stiff.

6

Gustafsson, Fundamentals of Scientific Computing

- λ

Even Euler method can be stable, if the

time step is not too large.

d u

_m

(t)

d t = D u

_m+1

(t )−2 u

_m

( t)+u

_m−1

( t) h

²

˙⃗u= ⃗f (⃗u) ⃗u

ⁿ⁺¹

=⃗u

ⁿ

+τ ⃗ f (⃗u

ⁿ

) u

_mⁿ⁺¹

=u

_mⁿ

+ D τ

h

²

(u

_m+1ⁿ

−2 u

_mⁿ

+u

_m−1ⁿ

) u

_m

(t)=e

^{ikhm−λ t}

.

For continuous time we had

u

_mⁿ

= e

^{ikhm−λ τ n}

e

ikhm−λ τ (n+1)

=e

^{ikhm−λ τ n}

+ D τ

h

²

( e

ikh(m+1)−λ τ n

− 2 e

^{ikhm−λ τ n}

+e

ikh( m−1)−λ τ n

)

e

^{−λ τ}

=1+ D τ

h

²

( e

^ikh

− 2+e

^−ikh

) ⇒ λ=− 1

τ ln [ ^{1− 2 D τ h}

²

^{{ 1−cos(kh)}} ]

Stability: Re λ ≥ 0 for all k.

∣ ^{1− 4 D h}

²

^{τ sin}

²

^{( kh/2)} ∣ ^{≤1 ⇒ τ≤ h 2 D}

²

=− 1

τ ln [ ^{1− 4 D h}

²

^{τ sin}

²

^{( kh/2) ] .}

Von Neumann stability analysis

Forward-time centred-space (FTCS) scheme 7

τ≤τ

_max

= h

²

2 D .

Stability criterion Rather stringent: quadratic in mesh step

O(h

²

)+O(τ)=O (h

²

)

Does not really make sense to use a higher-order explicit method for time evolution: the stability ranges of all of them are not very different, so τ ~ h² anyway, and then the time scheme is too accurate.

In fact, trying to be 2^nd order in time can make things worse. Richardson (1910) – leapfrog method.

∂ u( x , t)

∂ t = D ∂

²

u (x , t)

∂ x

²

Take centred difference in time.

u

_mⁿ⁺¹

− u

_mⁿ⁻¹

2 τ = D u

_m+1ⁿ

− 2 u

ⁿ_m

+u

_m−1ⁿ

h

²

u

_mⁿ

= e

^{ikhm−λ τ n}

e

^{−λ τ}

− e

^{λ τ}

2 τ = 2 D

h

²

(cos kh−1)

Denote

e

^{−λ τ}

≡G , b≡ 4 D τ

h

²

( 1−cos kh). G−1/G=−b ⇒ G=−b± √ ^b

²

^{+ 4}

2

For b≠0, one of the roots always has |G|>1 ⇒ λ < 0.

Unconditionally unstable!

8

giving a higher-order method.

λ=− 1

τ ln [ ^{1− 2 D τ h}

²

^{{ 1−cos(kh)}} ]

=− 1

τ ln [ ^{1− 2 D h}

²

^{τ {( kh)}

²

^{/ 2−(kh)}

⁴

^{/ 24+O ((kh)}

⁶

^{)} ]}

=2 D

h² {(kh)²/2−(kh)⁴/24+O((kh)⁶)}+ D²k⁴τ

2 =Dk²

[

^1+(kh)2

⁽

^{2 hD τ2−121}

⁾

^+O((kh)4)

]

τ= h

²

6 D = τ

_max

3 ,

When the method is O(h⁴).

M.V. Chubynsky and G.W. Slater, Phys. Rev. E 85, 016709.

On the other hand, going back to the FTCS scheme, since the error is

O(h

²

)+O (τ)

and τ has to be O(h²), these terms may cancel out 9

u

_mⁿ⁺¹

=u

_mⁿ

+ D τ

h

²

(u

_m+1ⁿ

( t )−2 u

ⁿ_m

( t)+u

_m−1ⁿ

( t))

For

τ=τ

_max

= h

²

2 D , u

m

n+1

= 1

2 ( u

_m+1ⁿ

+u

_m−1ⁿ

)

Suppose is 1 for even m and 0 for odd m. Then will be 0 for even m and 1 for odd m. Will oscillate forever. For larger τ this mode will grow.

u

_mⁿ

u

_mⁿ⁺¹

The small time step is needed to take care of short-wavelength modes. But they die down rapidly, so after a short while we don't care about them. But

cannot ignore them, because if the method is unstable, they will grow back. As we know for ODEs, the way to deal with that is implicit schemes that are

stable for arbitrarily large steps.

Boundary conditions are implemented as before: fix u on the boundary for Dirichlet and introduce ghost sites for Neumann. FTCS does conserve the particle number for reflecting BC or in infinite space.

10

u

_mⁿ⁺¹

− u

_mⁿ⁻¹

2 τ = D u

_m+1ⁿ

− 2 u

ⁿ_m

+u

_m−1ⁿ

h

²

Try to remove the instability by making it implicit:

u

_mⁿ⁺¹

−u

_mⁿ⁻¹

2 τ = D u

_m+1ⁿ

−(u

_mⁿ⁺¹

+u

ⁿ⁻¹_m

)+ u

_m−1ⁿ

h

²

A trick that does not really work:

Unconditionally unstable leapfrog

u

_mⁿ⁺¹

It turns out it is unconditionally stable. Moreover, even though it looks implicit, it's a linear equation for , so we can solve it. However,

u

_m+1ⁿ

−(u

_mⁿ⁺¹

+u

_mⁿ⁻¹

)+ u

_m−1ⁿ

h

²

= u

_m+1ⁿ

−2 u

_mⁿ

+u

_m−1ⁿ

h

²

− u

_mⁿ⁺¹

− 2 u

ⁿ_m

+u

_mⁿ⁻¹

h

²

→ ∂

²

u

∂ x

²

− τ

²

h

²

∂

²

u

∂ t

² So it is only a correct representation of the diffusion equation, when τ<<h, and we have not gained anything.

Dufort-Frankel scheme 11

d u

_m

( t)

d t = D u

_m+1

( t )−2 u

_m

( t)+u

_m−1

( t) h

²

Backward Euler:

˙⃗u= ⃗f (⃗u) ⃗u

ⁿ⁺¹

=⃗u

ⁿ

+τ ⃗ f ( ⃗ u

ⁿ⁺¹

) u

_mⁿ⁺¹

=u

_mⁿ

+ D τ

h

²

(u

_m+1ⁿ⁺¹

−2 u

_mⁿ⁺¹

+ u

_m−1ⁿ⁺¹

)

A tridiagonal set of equations for

u

_mⁿ⁺¹

Backward-time centred-space (BTCS) scheme.

1=e

^{λ τ}

+ D τ

h

²

( e

^ikh

− 2+e

^−ikh

) ⇒ λ= 1

τ ln [ ^{1+ 2 D τ h}

²

^{{ 1−cos(kh)}} ]

For FTCS we had:

λ=− 1

τ ln [ ^{1− 2 D τ h}

²

{1−cos(kh)} ]

λ ≥ 0 always. Unconditionally stable. But now 1^st order accuracy is an issue.

u

_mⁿ

= e

^{ikhm−λ τ n}

e

ikhm−λ τ (n+1)

=e

^{ikhm−λ τ n}

+ D τ

h

²

( ^e

ikh(m+1)−λ τ( n+1)

− 2 e

ikhm−λ τ (n+1)

+ e

ikh(m−1)−λ τ (n+1)

)

12

d u

_m

( t)

d t = D u

_m+1

( t )−2 u

_m

( t)+u

_m−1

( t) h

²

Trapezoidal method:

˙⃗u= ⃗f (⃗u) ⃗u

ⁿ⁺¹

=⃗u

ⁿ

+ τ

2 [ ⃗ f (⃗u

ⁿ

)+ ⃗ f (⃗u

ⁿ⁺¹

)]

u

_mⁿ⁺¹

=u

_mⁿ

+ D τ

2 h

²

( u

_m+1ⁿ⁺¹

− 2 u

_mⁿ⁺¹

+u

ⁿ⁺¹_m−1

+ u

_m+1ⁿ

− 2 u

_mⁿ

+u

ⁿ_m−1

)

Crank-Nicolson scheme (often misspelled Nicholson). 2^nd order in time as well

e

ikhm−λ τ (n+1)

=e

^{ikhm−λ τ n}

+ D τ

2 h

²

( e

ikh(m+1)−λ τ (n+1)

− 2 e

ikhm−λ τ (n+1)

+e

ikh(m−1)−λ τ (n+1)

)

+ D τ

2 h

²

( ^e

ikh(m+1)−λ τ n

− 2 e

^{ikhm−λ τ n}

+e

ikh(m−1)−λ τ n

)

e

^{−λ τ}

−1= D τ

h

²

(cos kh−1)(e

^{−λ τ}

+1) e

^{−λ τ}

≡G D τ

h

²

(1−cos kh)≡b

G=1 −b

1+b

|G| < 1 for b > 0. Unconditionally stable.

G−1=−b(G+1)

13

Recall a nice property of the trapezoidal scheme: if an ODE has a solution

with purely imaginary λ, the trapezoidal scheme preserves this property.

∼e

^λt

Useful for solving the time-dependent Schrödinger equation

Rescale t and x to get

i ∂ ψ

∂ t =− ∂

²

ψ

∂ x

²

+V ( x)ψ

If V(x) = 0 and there are no boundaries, we can still look for a solution of the form

ψ( x , t)=exp(−λ t+ikx). −i λ=k

²

⇒ λ=ik

²

.

The wave function preserves its norm; the evolution operator is unitary.

∫

∣ψ( x , t)∣

²

dx=const ψ( x ,t)=e

^{−i ̂H t}

ψ( x , 0)

Need

ψ

ⁿ⁺¹_j

= ̂ U ψ

ⁿ_j

, ̂ U

^H

U = ̂I ̂

^∫

(U ψ)

^∗

(U ψ)dx=

^∫

ψ

^∗

( U

^H

U ψ)dx=

^∫

ψ

^∗

ψ dx i ∂ ψ

∂ t = ̂ H ψ

14

Forward Euler method is unstable for purely imaginary λ. For FTCS,

ψ

_mⁿ⁺¹

=ψ

_mⁿ

−i τ ̂̃H ψ

_mⁿ

H ψ=− ̂ ∂

²

ψ

∂ x

²

+V (x) ψ

Discretized

̂̃H ψ

_mⁿ

=− ψ

_m+1ⁿ

−2 ψ

_mⁿ

+ψ

ⁿ_m−1

h

²

+V

_m

ψ

_mⁿ

ψ

_mⁿ⁺¹

=(1−i τ ̂̃ H )ψ

_mⁿ

∂ ψ

∂ t =−i ( ^{− ∂ ∂}

²

^{x ψ}

²

^{+V ( x)ψ} ) ^{=−i ̂H ψ}

(1−i τ ̂̃H )

^H

(1−i τ ̂̃ H )=(1+i τ ̂̃H )(1−i τ ̂̃ H )=1+τ

²

̂̃H

²

≠1

In Backward Euler method modes with purely imaginary λ decay. For BTCS,

ψ

_mⁿ⁺¹

=ψ

_mⁿ

−i τ ̂̃H ψ

_mⁿ⁺¹

ψ

_mⁿ⁺¹

=( 1+i τ ̂̃ H )

⁻¹

ψ

_mⁿ

(1+i τ ̂̃H )

^−H

( 1+i τ ̂̃H )

⁻¹

=[(1+i τ ̂̃H )(1−i τ ̂̃ H )]

⁻¹

=(1+τ

²

̂̃H

²

)

⁻¹

≠1

15

In the trapezoidal method, the oscillations do not grow nor decay.

H ψ=− ̂ ∂

²

ψ

∂ x

²

+V (x) ψ

Discretized

̂̃H ψ

_mⁿ

=− ψ

_m+1ⁿ

−2 ψ

_mⁿ

+ψ

ⁿ_m−1

h

²

+V

_m

ψ

_mⁿ

∂ ψ

∂ t =−i ( ^{− ∂ ∂}

²

^{x ψ}

²

^{+V ( x)ψ} ) ^{=−i ̂H ψ}

For Crank-Nicolson,

ψ

_mⁿ⁺¹

=ψ

_mⁿ

−( i τ ̂̃H /2)(ψ

_mⁿ

+ψ

_mⁿ⁺¹

) (1+i τ ̂̃H /2) ψ

_mⁿ⁺¹

=(1−i τ ̂̃ H /2)ψ

_mⁿ

ψ

_mⁿ⁺¹

=(1+i τ ̂̃ H /2)

⁻¹

( 1−i τ ̂̃H /2) ψ

_mⁿ

[(1+i τ ̂̃ H /2)

⁻¹

(1−i τ ̂̃H /2)]

^H

(1+i τ ̂̃ H /2)

⁻¹

(1−i τ ̂̃H /2)=

(1−i τ ̂̃ H /2)

⁻¹

(1+i τ ̂̃H /2)(1+i τ ̂̃ H /2)

⁻¹

(1−i τ ̂̃H /2)=1

Can also see from von Neumann analysis. We had

e

^{−λ τ}

≡G G=1 −b

1+b b → ib ∣ ^{1−ib 1+ib} ∣ ⁼¹

16

Another, explicit possibility is, surprisingly, the leapfrog method

ψ

_mⁿ⁺¹

−ψ

_mⁿ⁻¹

2 τ =−i ̂̃H ψ

_mⁿ

⇒ ψ

_mⁿ⁺¹

=ψ

_mⁿ⁻¹

−2 i τ ̂̃ H ψ

_mⁿ

e

^{−λ τ}

≡G , b≡ 4 D τ

h

²

(1−cos kh). G= −b± √ ^b

²

^{+ 4}

2 b →ib ⇒ G= − i∣b∣± √ ^{−∣ b∣}

²

^{+ 4}

2 ^{∣ G∣=}

√ ^4−∣b∣

²

^{+∣ b∣}

²

2 =1 , ∣b∣<2

Need

ψ

¹_m

= e

^{−i τ ̂H}

ψ

⁰_m

≈

^∑2_j=0

(−i τ ̂̃ H )

^j

j ! ψ

⁰_m 17

Higher dimensions

Of course, the explicit methods (FTCS for diffusion eq. and leapfrog for Schrodinger eq.) can be generalized straightforwardly.

u

ⁿ⁺¹_{j ,l}

=u

ⁿ_{j ,l}

+ D τ

h

²

(u

ⁿ_{j+1, l}

+u

ⁿ_j−1,l

+ u

ⁿ_{j ,l +1}

+u

ⁿ_{j , l−1}

− 4 u

ⁿ_{j ,l}

)+τ D f

_{j , l}

In this case,

∂ u( x , y , t)

∂ t = D ( ^∂

²

u (x , y , t)

∂ x

²

+ ∂

²

u (x , y , t)

∂ y

²

) ⁺ D f ( x , y)

τ

_max

=h

²

/ 4 D u

ⁿ⁺¹_{j ,l}

= 1

4 ( u

ⁿ_j+1,l

+u

ⁿ_{j−1, l}

+u

ⁿ_{j , l+1}

+u

ⁿ_{j ,l −1}

)+ h

²

4 f

_{j ,l}

This is exactly the Jacobi iteration for the elliptic problem

( ^∂

²

u (x , y ,t)

∂ x

²

+ ∂

²

u(x , y ,t)

∂ y

²

) ^{=− f (x , y)}

18

Higher dimensions

It is possible in principle to use the Crank-Nicolson method in higher dimensions, but the matrix will not be tridiagonal, but rather will have a structure similar to that we encountered for elliptic equations.

Alternating direction implicit (ADI) method: split each step into 2 substeps, treating 1 direction implicitly and the other explicitly in each substep.

u

^{n+1/ 2}_{j ,l}

=u

ⁿ_{j , l}

+ D τ

2 h

²

( u

^n+1/2_{j+1, l}

− 2 u

^{n+1/ 2}_{j ,l}

+ u

^n+1/2_j−1,l

+u

ⁿ_{j ,l +1}

− 2 u

ⁿ_{j , l}

+u

ⁿ_{j ,l −1}

) u

ⁿ⁺¹_{j ,l}

=u

ⁿ_{j ,l}

+ D τ

2 h

²

(u

^{n+1 /2}_j+1,l

− 2 u

^{n+1/ 2}_{j , l}

+u

^{n+1/ 2}_j−1,l

+u

ⁿ⁺¹_{j , l+1}

− 2 u

ⁿ⁺¹_{j , l}

+u

ⁿ⁺¹_{j , l−1}

)

A particular case of general operator splitting methodology (if there are several terms, each can be treated in a separate substep), although with a slightly different twist.

19

Nonlinear equations

Straightforward with explicit methods. Hard with implicit methods. Lagging nonlinear coefficients, iteration, Newton's method.

20