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Example 3.7a
Consider the convective diffusion problem (Finlayson,1980)[11]
(3.65)
An analytical solution can be obtained using the exponential matrix method
described in section
3.1.2:
This particular problem was chosen as the finite difference solution for this
equation and shows
oscillations for high Peclet numbers when the central difference expression
is used for the first
derivative. This equation is solved below using the procedure described
above.
> restart:
> with(linalg):with(plots):
> N:=4;
(1)
> L:=1;
(2)
> eq:=diff(y(x),x$2)-Pe*diff(y(x),x);
(3)
> bc1:=y(x)-1;
(4)
> bc2:=y(x);
(5)
Central difference expressions for the second and first derivatives are
> d2ydx2:=(y[m+1]-2*y[m]+y[m-1])/h^2;
(6)
> dydx:=(y[m+1]-y[m-1])/2/h;
(7)
The governing equation in finite difference form is:
> Eq[m]:=subs(diff(y(x),x$2)=d2ydx2,diff(y(x),x)=dydx,y(x)=y[m],x=m*h,eq);
(8)
A 'for loop' can be written for the interior node points as
> for i to N do Eq[i]:=subs(m=i,Eq[m]);od;
(9)
> Eq[0]:=y[0]=1;
(10)
> Eq[N+1]:=y[N+1]=0;
(11)
> y[0]:=solve(Eq[0],y[0]);
(12)
> y[N+1]:=solve(Eq[N+1],y[N+1]);
(13)
> h:=L/(N+1);
(14)
> for i to N do Eq[i]:=eval(Eq[i]);od;
(15)
> eqs:=[seq(Eq[i],i=1..N)];
(16)
> vars:=[seq(y[i],i=1..N)];
(17)
> A:=genmatrix(eqs,vars,'B1');
(18)
> evalm(B1);
(19)
Maple generates a row vector, which can be converted to a column vector as:
> B:=matrix(N,1):for i to N do B[i,1]:=B1[i]:od:evalm(B);
(20)
The solution is obtained as:
> X:=evalm(inverse(A)&*B);
(21)
> for i to N do y[i]:=X[i,1];od;
(22)
> y[0]:=eval(y[0]);y[N+1]:=eval(y[N+1]);
(23)
Next, the result obtained is compared with the exact analytical solution:
> ya:=(exp(Pe)-exp(Pe*x))/(exp(Pe)-1);
(24)
>
p1:=plot([seq([i*h,subs(Pe=1,y[i])],i=0..N+1)],thickness=4,color=blue,axes=bo
xed):
>
p2:=plot(subs(Pe=1,ya),x=0..1,thickness=8,color=brown,axes=boxed,linestyle=2)
:
> display({p1,p2},title="Figure Exp. 3.1.9.",labels=[x,"y"]);
We observe that both the finite difference solution and the analytical
solution match exactly
when the Peclet number is 1. New plots can be obtained for different values
of the Peclet
number as follows:
>
p1:=plot([seq([i*h,subs(Pe=50,y[i])],i=0..N+1)],color=blue,thickness=4,axes=b
oxed):
>
p2:=plot(subs(Pe=50,ya),x=0..1,thickness=5,color=brown,axes=boxed,linestyle=2
):
> display({p1,p2},title="Figure Exp. 3.1.10.",labels=[x,"y"]);
This shows that for Pe = 50, four interior node points are not enough and we
observe
oscillations.[11][12] This happens usually when central difference
approximations are used for the
convective term
. Use a forward approximation for the first derivative to solve this
problem. Only dydx in the Maple program needs to be changed:
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