# Solution Instability Explicit Method

**Text the Difference Equations in Matrix Form**:

If we use the boundary conditions u(0) = u(L) = 0 then the explicit method of the previous section has the form:

u_{i,j+1} = ru_{i−1,j }+ (1 − 2r)u_{i,j} + ru_{i+1,j} , 1 ≤ i ≤ m − 1, 0 ≤ j ≤ n − 1,

where u_{0,j} = 0 and u_{m,j}= 0. This is equal to the matrix equation:

u_{j+1} = Au_{j},

where u_{j} is the column vector (u_{1,j}, u_{2,j}, . . . , u_{m,j})′ representing the state at the jth time step and A is the matrix:

Unluckily this matrix can have a property which is extremely bad in this context. Specifically it can cause exponential growth of error unless r is small. To observe how this happens suppose that Ujis the vector of correct values of u at time step tjand Ejis the error of the approximation uj

u_{j}= U_{j}+ E_{j}.

The estimation at the next time step will be:

u_{j+1} = AU_{j}+ AE_{j},

And if we continue for k steps,

u_{j+k}= AkU_{j}+ AkE_{j}.

The problem with this is the term AkE_{j}. This term is precisely what we would do in the power method for finding the eigenvalue of A with the largest absolute value. If the matrix A has ew’s with total value greater than 1 then this term will grow exponentially. The largest complete value of an ew of A as a function of the parameter r for various sizes of the matrix.

A. As you can observe for r >1/2 the largest absolute ew grows rapidly for any m and rapidly becomes greater than 1.

Maximum complete eigenvalue EW as a function of r for the matrix A from the explicit method for the heat equation calculated for matrices A of sizes m = 2 . . . 10. When EW >1 the method is unstable that is errors grow exponentially with each step. When by means of the explicit method r <1/2 is a safe choice.

**Consequences**:

Recall that r = ck/h^{2}. Since this should be less than the 1/2, so we have:

k <h^{2}/2c.

The first consequence is observable k should be relatively small. The second is that h can’t be too small. Ever since h^{2} appears in the formula making h small would force k to be extremely small!

A third consequence is that we comprise a converse of this analysis. Presume r < .5. Then every eigenvalues will be less than one. Evoke that the error terms satisfy:

u_{j+k}= AkU_{j}+ AkE_{j}.

If all the eigenvalues of A are less than 1 in complete value then AkE_{j} grows smaller and smaller as k increases. This is actually good. Rather than building up the consequence of any error diminishes as time passes! From this we get there at the following principle- If the explicit numerical solution for a parabolic equation doesn’t blow up then errors from previous steps fade away!

Ultimately we note that if we have other boundary conditions then instead of equation we have:

u_{j+1} = Au_{j}+ rb_{j}

Where the first as well as last entries of bjcontain the boundary conditions and all the other entries are zero. In this case the errors perform just as before if r >1/2 then the errors grow and if r <1/2 the errors fade away.

We are able to write a function program myexppmatrix that produces the matrix A in for given inputs m and r. Without using loops we are able to use the diag command to set up the matrix

*function A = myexpmatrix(m,r)*

*% produces the matrix for explicit method for a parabolic equation*

*% Inputs: m -- the size of the matrix*

*% r -- the main parameter, ck/h^2*

*% Output: A -- an m by m matrix*

*u = (1-2*r)*ones(m,1);*

*v = r*ones(m-1,1);*

*A = diag(u) + diag(v,1) + diag(v,-1);*

Test this by means of m = 6 and r = .4.6, Check the eigenvalues as well as eigenvectors of the resulting matirices

> A = myexpmatrix

> [v e] = eig(A)

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