Scientific Computing

Partial Differential EquationsImplicit Solution of

Heat Equation

Explicit vs Implicit Methods

• Explicit methods have problems relating to stability

• Implicit methods overcome this but at the expense of introducing a more complicated algorithm

• In the implicit algorithm, we develop simultaneous equations for u at the j-th and (j+1)-st time steps

Explicit Method

We use the centered-difference approximation for uxx at time step j:

€

ui−1, j − 2ui, j + ui+1, j

h2

grid point involved with space difference grid point involved with time difference

Implicit Method

We use the centered-difference approximation for uxx at step (j+1) :

€

ui−1, j+1 − 2ui, j+1 + ui+1, j+1

h2

grid point involved with space difference grid point involved with time difference

• We use the forward-difference formula for the time derivative and the centered-difference formula for the space derivative at time step (j+1):

Implicit Method

€

ut (x i, t j ) ≅ui, j+1 − ui, j

k

cuxx (x i, t j ) ≅ cui−1, j+1 − 2ui, j+1 + ui+1, j+1

h2

• Then the heat equation (ut =cuxx ) can be approximated as

Or,

Let r = (ck/h2) Solving for ui,j we get:

Implicit Method

€

ui, j+1 − ui, jk

= cui−1, j+1 − 2ui, j+1 + ui+1, j+1

h2

€

ui, j+1 − uij =ck

h2 ui−1, j+1 − 2ui, j+1 + ui+1, j+1( )

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ui, j = −rui−1, j+1 + (1+ 2r)ui, j+1 − rui+1, j+1( )

• Putting in the boundary conditions, we get the following equations for the (implicit) solution to the heat equation is:

Implicit Method

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ui, j = −rui−1, j+1 + (1+ 2r)ui, j+1 − rui+1, j+1( ) j ≥ 0 i = 2,K n − 2

u1, j = −rg0, j+1 + (1+ 2r)u1, j+1 − ru2, j+1( ) j ≥ 0

un−1, j = −run−2, j+1 + (1+ 2r)un−1, j+1 − rg1, j+1( ) j ≥ 0

ui,0 = f i i = 0K n

Matrix Form of Solution

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1+ 2r −r

−r 1+ 2r −r

O O O

−r 1+ 2r −r

−r 1+ 2r

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

u1, j+1

u2, j+1

M

un−2, j+1

un−1, j+1

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

+

−rg0, j+1

0

M

0

−rg1, j+1

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

=

u1, j

u2, j

M

un−2, j

un−1, j

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

€

ui, j = −rui−1, j+1 + (1+ 2r)ui, j+1 − rui+1, j+1( ) j ≥ 0 i = 2,K n − 2

u1, j = −rg0, j+1 + (1+ 2r)u1, j+1 − ru2, j+1( ) j ≥ 0

un−1, j = −run−2, j+1 + (1+ 2r)un−1, j+1 − rg1, j+1( ) j ≥ 0

ui,0 = f i i = 0K n

This is of the formTo solve for u(:,j) we need to use a linear systems method.Since we have to solve this system repeatedly, a good choice

is the LU decomposition method.

Matrix Form of Solution

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Au(:, j +1) +b = u(:, j)

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1+ 2r −r

−r 1+ 2r −r

O O O

−r 1+ 2r −r

−r 1+ 2r

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

u1, j+1

u2, j+1

M

un−2, j+1

un−1, j+1

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

+

−rg0, j+1

0

M

0

−rg1, j+1

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

=

u1, j

u2, j

M

un−2, j

un−1, j

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

function z = implicitHeat(f, g0, g1, T, n, m, c)%Simple Implicit solution of heat equation % Constants h = 1/n; k = T/m; r = c*k/h^2; % x and t vectors x = 0:h:1; t = 0:k:T; % Boundary conditions u(1:n+1, 1) = f(x)'; u(1, 1:m+1) = g0(t); u(n+1, 1:m+1) = g1(t);

Matlab Implementation

% Set up tri-diagonal matrix for interior points of the grid A = zeros(n-1, n-1); % 2 less than x grid size for i= 1: n-1 A(i,i)= (1+2*r); if i ~= n-1 A(i, i+1) = -r; end if i ~= 1 A(i, i-1) = -r; end end % Find LU decomposition of A [LL UU] = lu_gauss(A); % function included below

Matlab Implementation

% Solve for u(:, j+1); for j = 1:m % Set up vector b to solve for in Ax=b b = zeros(n-1,1); b = u(2:n, j)'; % Make a column vector for LU solver b(1) = b(1) + r*u(1,j+1); b(n-1) = b(n-1) + r*u(n+1,j+1); u(2:n, j+1) = luSolve(LL, UU, b); end z=u'; % plot solution in 3-d mesh(x,t,z);end

Matlab Implementation

Usage: f = inline(‘x.^4’); g0 = inline(‘0*t’); g1 = inline(‘t.^0’); n=5; m=5; c=1; T=0.5; z = implicitHeat(f, g0, g1, T, n, m, c);

Matlab Implementation

Calculated solution appears stable:

Matlab Implementation

To analyze the stability of the method, we again have to consider the eigenvalues of A for the equation

We have

Using the Gershgorin Theorem, we see that eigenvalues are contained in circles centered at (1+2r) with max radius of 2r

Stability

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Au(:, j +1) +b = u(:, j)

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A =

1+ 2r −r

−r 1+ 2r −r

O O O

−r 1+ 2r −r

−r 1+ 2r

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Thus, if λ is an eigenvalue of A, we have

Thus, all eigenvalues are at least 1. In solving for u(:,j+1) in we haveSince the eigenvalues of A-1 are the reciprocal of the

eigenvalues of A, we have that the eigenvalues of A-1 areall <= 1. Thus, the implicit algorithm yields stable iterates – they do

not grow without bound.

Stability

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1+ 2r − 2r ≤ λ ≤1+ 2r + 2r

⇒ 1≤ λ ≤1+ 4r

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Au(:, j +1) +b = u(:, j)

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u(:, j +1) = A−1(u(:, j) −b)

• Convergence means that as x and t approach zero, the results of the numerical technique approach the true solution

• Stability means that the errors at any stage of the computation are attenuated, not amplified, as the computation progresses

• Truncation Error refers to the error generated in the solution by using the finite difference formulas for the derivatives.

Convergence and Stability

Example: For the explicit method, it will be stable if r<= 0.5 and will have truncation error of O(k+h2) .

Implicit Method is stable for any choice of r and will again have truncation error of O(k+h2) .

It would be nice to have a method that is O(k2+h2). The Crank-Nicholson method has this nice feature.

Convergence and Stability

Crank-Nicholson Method

We average the centered-difference approximation

for uxx at time steps j and j+1:

€

ui−1, j − 2ui, j + ui+1, j

h2

grid point involved with space difference grid point involved with time difference

€

ui−1, j+1 − 2ui, j+1 + ui+1, j+1

h2

Crank-Nicholson Method

We get the following

€

ui, j+1 − uij =ck

2h2 ui−1, j − 2ui, j + ui+1, j( )

+ck

2h2 ui−1, j+1 − 2ui, j+1 + ui+1, j+1( )

• Putting in the boundary conditions, we get the following equations for the (C-N) solution to the heat equation is:

Crank-Nicholson Method

€

−r

2ui−1, j+1 + (1+ r)ui, j+1 −

r

2ui+1, j+1

⎛

⎝ ⎜

⎞

⎠ ⎟=r

2ui−1, j + (1− r)ui, j +

r

2ui+1, j

⎛

⎝ ⎜

⎞

⎠ ⎟

−r

2g0, j+1 + (1+ r)u1, j+1 −

r

2u2, j+1

⎛

⎝ ⎜

⎞

⎠ ⎟=r

2g0, j + (1− r)u1, j +

r

2u2, j

⎛

⎝ ⎜

⎞

⎠ ⎟

−r

2un−2, j+1 + (1+ r)un−1, j+1 −

r

2g1, j+1

⎛

⎝ ⎜

⎞

⎠ ⎟=r

2un−2, j + (1− r)un−1, j +

r

2g1, j

⎛

⎝ ⎜

⎞

⎠ ⎟

ui,0 = f i

• Class Project: Determine the matrix for the C-N method and revise the Matlab implicitHeat function to implement C-N.

Crank-Nicholson Method

€

−r

2ui−1, j+1 + (1+ r)ui, j+1 −

r

2ui+1, j+1

⎛

⎝ ⎜

⎞

⎠ ⎟=r

2ui−1, j + (1− r)ui, j +

r

2ui+1, j

⎛

⎝ ⎜

⎞

⎠ ⎟

−r

2g0, j+1 + (1+ r)u1, j+1 −

r

2u2, j+1

⎛

⎝ ⎜

⎞

⎠ ⎟=r

2g0, j + (1− r)u1, j +

r

2u2, j

⎛

⎝ ⎜

⎞

⎠ ⎟

−r

2un−2, j+1 + (1+ r)un−1, j+1 −

r

2g1, j+1

⎛

⎝ ⎜

⎞

⎠ ⎟=r

2un−2, j + (1− r)un−1, j +

r

2g1, j

⎛

⎝ ⎜

⎞

⎠ ⎟

ui,0 = f i