Discrete Maximum Principle

∂V

∂t− σ2

2

∂2V

∂x2= 0

1 Explicit Scheme:

V n+1j − V n

j

∆t− σ2

2

V nj+1 + V n

j−1 − 2V nj

∆x2= 0

or

V n+1j =

σ2

2

∆t

∆x2V n

j+1 +σ2

2

∆t

∆x2V n

j−1 +(1− σ2 ∆t

∆x2

)V n

j .

If,

1− σ2 ∆t

∆x2≥ 0. (1)

Then ∥∥∥V n+1∥∥∥ ≤ ‖V n‖ (2)

where ‖V n‖ =maxi|V n

i | . In essence, (2) means the maximum principle, where

the condition (1) plays a critical role.Let u(x, t) be the true solution.

un+1j =

σ2

2

∆t

∆x2un

j+1 +σ2

2

∆t

∆x2un

j−1 +(1− σ2 ∆t

∆x2

)un

j + O(∆t2 + ∆t∆x2

).

Denotezn

j = unj − V n

j .

We have

zn+1j =

σ2

2

∆t

∆x2zn

j+1 +σ2

2

∆t

∆x2zn

j−1 +(1− σ2 ∆t

∆x2

)zn

j + O(∆t2 + ∆t∆x2

).

1

Then ∥∥∥zn+1∥∥∥ ≤ ‖zn‖+ C

(∆t2 + ∆t∆x2

).

So

‖zn‖ ≤ nC(∆t2 + ∆t∆x2

)

≤ T

∆tC

(∆t2 + ∆t∆x2

)

= TC(∆t + ∆x2

),

which implies the convergence.

2 Fully Implicit Scheme

V n+1j − V n

j

∆t− σ2

2

V n+1j+1 + V n+1

j−1 − 2V n+1j

∆x2= 0.

Proposition 1 Discrete Maximum Principle (comparison princi-ple): If

V n+1j − V n

j

∆t− σ2

2

V n+1j+1 + V n+1

j−1 − 2V n+1j

∆x2≤ 0

andV n

j ≤ 0 on parabolic boundary,

then V nj ≤ 0 for all n and j.

Proof: Assume it is true at the n−th step. Suppose V n+1j is a local

maximum inside the domain. Then

0 ≤ V n+1j − V n

j

∆t− σ2

2

V n+1j+1 + V n+1

j−1 − 2V n+1j

∆x2≤ 0.

It followsV n

j = V n+1j+1 = V n+1

j−1 = V n+1j .

This implies that the maximum must occur on the parabolic boundary. SoV n

j ≤ 0 for all n and j.

Let u be the true solution and znj = un

j − V nj . We have

2

zn+1j − zn

j

∆t− σ2

2

zn+1j+1 + zn+1

j−1 − 2zn+1j

∆x2= O

(∆t + ∆x2

).

We construct a function (on mesh points)

wnj = zn

j ± C(∆t + ∆x2

)n∆t.

Then

wn+1j − wn

j

∆t− σ2

2

wn+1j+1 + wn+1

j−1 − 2wn+1j

∆x2

=zn+1

j − znj

∆t− σ2

2

zn+1j+1 + zn+1

j−1 − 2zn+1j

∆x2± C

(∆t + ∆x2

)

= O(∆t + ∆x2

)± C

(∆t + ∆x2

).

So,‖zn‖ ≤ C

(∆t + ∆x2

)n∆t = CT

(∆t + ∆x2

).

3 Crank-Nicolson Scheme

V n+1j − V n

j

∆t− σ2

4

[V n+1

j+1 + V n+1j−1 − 2V n+1

j

∆x2+

V nj+1 + V n

j−1 − 2V nj

∆x2

]= 0.

Using Fourier analysis, it can be shown that the scheme is unconditionallystable and thus is always convergent. (See Thomas (1995): Numerical partialdifferential equations: finite difference methods. Springer-Verlag.)

However, if we make use of the discrete maximum principle to show theconvergence, we will have to add the condition

1− σ2

2

∆t

∆x2≥ 0. (3)

I leave this as an exercise.

3