Finite-Difference Approximations to the HeatEquation

From notes by Gerald W. Recktenwald

Finite-Difference Approximations to the Heat Equation

The Heat Equation

∂φ

∂t= α

∂2φ

∂x2

with 0 ≤ x ≤ L, t ≥ 0.

We assume that we are given

φ(0, t) = φ0, φ(L, t) = φL, φ(x , 0) = f0(x)

Finite-Difference Approximations to the Heat Equation

Finite Difference Method

We replace all he derivatives by discrete approximations.

In the heat equation there are derivatives wrt time and space.

We need to choose how we approximate the derivatives usingpoints on the mesh.

Then, convergence has to be addressed (as ∆x and ∆t go to0).

Finite-Difference Approximations to the Heat Equation

Finite Difference Method

Finite-Difference Approximations to the Heat Equation

Mesh

We divide the space as follows:

xi = (i − 1)∆x where 1 ≤ i ≤ N (for 0 ≤ x ≤ L).tm = (m − 1)∆t where 1 ≤ m ≤ M (for 0 ≤ t ≤ tmax).

Finite-Difference Approximations to the Heat Equation

Mesh

Finite-Difference Approximations to the Heat Equation

Mesh

Finite-Difference Approximations to the Heat Equation

Finite Difference Approximations

Notation: φi = φ(xi ), φi+1 = φ(xi + ∆x).

First Order Forward Approximation:

∂φ

∂x

∣∣∣∣xi

≈ φi+1 − φi∆x

+∆x2

2

∂2φ

∂x2

∣∣∣∣ξ

Also, we can write

∂φ

∂x

∣∣∣∣xi

=φi+1 − φi

∆x+ O(∆x)

First Order Backward Approximation:

∂φ

∂x

∣∣∣∣xi

=φi − φi−1

∆x+ O(∆x)

Finite-Difference Approximations to the Heat Equation

Finite Difference Approximations

First Order Central Approximation:

∂2φ

∂x2

∣∣∣∣xi

=φi+1 − φi−1

2∆x+ O(∆x2)

Second Order Central Approximation:

∂φ

∂x

∣∣∣∣xi

=φi+1 − 2φi + φi−1

∆x2+ O(∆x2)

Finite-Difference Approximations to the Heat Equation

Schemes for the Heat Equation. FTCS

∂φ

∂t

∣∣∣∣tm,xi

=φm+1i − φmi

∆t+ O(∆t)

Then, combining it with the second order, central approximationfor x :

φm+1i − φm+1i

∆t= α

φmi+1 − 2φmi + φmi−1∆x2

+ O(∆x2) + O(∆t)

Then, we can solve for φm+1 (advance in time):

φm+1i = φmi + α∆t

φmi+1 − 2φmi + φmi−1∆x2

+ O(∆x2) + O(∆t)

Finite-Difference Approximations to the Heat Equation

Schemes for the Heat Equation. FTCS

Notice that the system we obtained is easy to solve since we cansolve for the φm+1s one at the time(explicit).

However, the obtained solutions (to the discrete system, soapproximations to the real solutions) can be unstable unlessr = α∆t∆x <

12 .

Solving that system is equivalent to solving φ(m+1) = Aφ(m) where:

A =

1 0 0 0 0 0r (1 − 2r) r 0 0 00 r (1 − 2r) r 0 0

. . . . . . . . . . . . . . . . . .0 0 0 r (1 − 2r) r0 0 0 0 0 1

Finite-Difference Approximations to the Heat Equation

Schemes for the Heat Equation. BTCS

If we use the backward approximation for the t derivative:

∂φ

∂t

∣∣∣∣tm,xi

=φmi − φ

m−1i

∆t+ O(∆t)

We obtain a different scheme.

It is more complicated to solve.

We can’t solve one-by-one.

These type of schemes is called implicit.

Finite-Difference Approximations to the Heat Equation

Schemes for the Heat Equation. Crank-Nicolson

The time-truncation in both schemes turns out to be O(∆t).

Combining the derivatives at the current and previous times for ximproves it to O(∆t2).

φmi − φm−1i

∆t= α

1

2(φmi+1 − 2φmi + φmi−1

∆x2+φm−1i+1 − 2φ

m−1i + φ

m−1i−1

∆x2)

Finite-Difference Approximations to the Heat Equation

Error Orders

Finite-Difference Approximations to the Heat Equation

Stencils

Finite-Difference Approximations to the Heat Equation

Stencils

Finite-Difference Approximations to the Heat Equation

FTCS

Finite-Difference Approximations to the Heat Equation

BTCS

Finite-Difference Approximations to the Heat Equation