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☰ Search Explore Log in Create new account Upload × 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: Download 1. Math 2. Algebra Example 3.7a.doc Example3.7a rev 1.docx Example 3.7b.doc Document Example2.2.1.doc Math 252 Applied Linear Algebra 1 Practice Test Group 4C with solution Chapter 3 THE RANK OF THE 2ND GAUSSIAN MAP FOR GENERAL CURVES Introduction Example3.2.3a Rev 1.docx Example3.2.3 Rev 1.docx An introduction to Coding Theory Visualizing vectors in 2D MATH 2270-2 Symmetric matrices, conics and quadrics MATH 2270-2 Symmetric matrices and quadratics Math 4530 Friday March 28 first and second fundamental forms in Maple Math 4530 Friday April 20 Christoffel symbols and Gauss’ Theorem Egregium > restart: studylib © 2017 DMCA Report