08 solution heat equation explicit

8/6/2019 08 Solution Heat Equation Explicit

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Heat EquationSolution sheet6. week

8 Heat Equation - Explicit

Aim of this exercise is the numerical solution of the heat equation (seeexercise 7) in explicit form.

a) Simplify the heat equation from exercise 7 to the one dimensional case.

Solution:t k

2x = 0 k > 0

b) Discretize the time derivative operator as usual (Euler-method).

Solution:

(x)n+1 = (x)n + t k 2x(x)

c) Discretize the spatial derivative in explicit form. The second derivativeis discretized in two steps:

First discretize the outer derivative as the divergence of theflux. The resulting scalar quantity is therefore stored at the cellcenter.

Secondly discretize the inner spatial derivative x at cell i.The result is assumed to be the above mentioned flux (in 3D thegradient is indeed a vector quantity) and should therefore bestored at the cell interfaces (staggered mesh).

Solution:

The outer derivative of F(x) = x yields to

(xF(x))i Fi+1/2 Fi1/2

x

The inner derivative yields to

Fi1/2 = (x)i1/2 i i1

x

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Combining these

2x(x) = x(x)

Fi+1/2 Fi1/2

x

(i+1 i) (i i1)

x2

=i+1 2i + i1

x2

The discretized heat equation in explicit form is

n+1i = ni + t kn

i+1 2n

i+ n

i1x2

d) Every explicit method yields to an upper limit of the timestep tmax.Derive this upper limit analytically (and test your result later on).

Solution:

The CFL condition leads to tmax = minxi|vi|

. In this particular

case the grid spacing x is constant in time and space and the velocityis given by the diffusion velocity:

F = vdiff = k

vdiff = kx(x)

(x)

The resulting upper limit for the timestep is

tmax = min

(x)x

k|x(x)|

e) Solve the explicit form of the heat equation numerically:

Use 100 grid points, x = 1.

Initial condition (for the density distribution) should be a simpleparabolic function (ax2 + b) with (x = xmin, t = 0) = 1 and

(x = xmax, t = 0) = 0.

The boundary of the domain is assumed to be an empty reservoirwith infinite capacity (x = ghost, t) = 0.

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Determine the upper limit of the timestep for this specific problem.

Solution:

With x(x) (x)x the upper limit of the timestep for this specific

problem is given by

tmax = min

(x)x2

k|(x)|

The highest jump in density is given at the left border. The density atthis innermost interface is 1/2 = 0.5(ghost + 1) = 0.5 and the jumpin density 1/2 = 1 ghost = 1. Therefore this scheme is unstable

for t > 0.5x2

k .Download the corresponding Mathematica file.

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08 solution heat equation explicit

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