EPDE Practice Problems

MCS 571 --- EPDE Practice Problems --- Hanson
Numerical Methods for PDEs

Instructions: In all computations, use 4 digit ``exam precision'': Chop all recorded intermediate results to 4 significant digits and continue calculations with only these chopped digits. Also summarize your results in a table for each U_{i,j} or U_{i,j}^{(k)}, if the problem is computational.
CAUTION: This collection contains some old and some recent problems, so that the former may correspond to different methods or text than those currently used and answers may not be the same as those by more recent methods.

Problems:

Solve Poisson's equation,
u_{xx} + u_{yy} = -x,
numerically on the 6 X 6 square, subject to the BC, u(0,y)=15y, u(x,0)=5x, u(6,y)=60, u(x,6)=120, and with
(Cautionary note: this problem was based on Gerald & Wheatley's notation, so that (1+2 \rho) is coefficient of diagonal term in both x & y ADI equations here) for one full iterate in both x and y directions.
Use the PRADI (Peaceman-Rachford or usual ADI) method to numerically approximate the solution of the EPDE problem,
2 u_{xx} + 3 u_{yy} + 0.1 x u_x + 1 = 0,
on 0 < x < 1.5, 0 < y < 1.5, with u(0,y)=0, u(x,0)=0, u(1.5,y)=3 y and u(x,1.5)=6 x. Take discrete steps h_1=0.5 and h_2=0.5. Let \rho=2.0 and U_{i,j}^{(1)}=0.5 as the starting iterate for interior (x_i,y_j). Iterate just until the next full iterate is obtained.

Use the SOR method with \omega=1.3 on the above EPDE PRADI problem for two full iterates beyond the initial one.

(1-\omega)

U_{i,j}^{(k)}

    k   U_{2,2}^{(k)}   U_{3,2}^{(k)}   U_{2,3}^{(k)}   U_{3,3}^{(k)}
    0      .5000           .5000           .5000           .5000
    1      .2083           .5258           1.264           3.541
    2      .6005           1.804           1.983           3.312

Use the SOR method with \omega=1.25 (where the normalization is so that (1-\omega) is the coefficient of the principal diagonal term on the RHS of the SOR difference equation) to numerically approximate the solution of
a u_{xx} + b u_{yy} + c u_x + d u_y + e u = 0,
where a=1.2, b=0.8, c=0.7 x, d=0.5 y and e=-2.0; u(x,0)=x/2, u(0,y)=y/4, u(1,y)=(2+y)/2 and u(x,1)=(x+0.5)/2; on 0 < x < 1, 0 < y < 1. Finite difference with h_1=0.25=h_2 and with initial iterate
U_{i,j}^{(0)} = x_i/2 + y_j/4.
Compute two (2) full iterates beyond this initial iterate. Verify that the coefficient matrix of the Jacobi method is diagonally dominant (i.e. ||D||_{\infty} > ||L + U||_{\infty}, assuming the \infty-norm or max. row sum norm).
Consider the point Jacobi Method with central finite differences for the constant coefficient EPDE,
\sum_{j=1}^n {a_j u_{x_jx_j} + b_j u_{x_j}} +c u + f = 0,
in n dimensions on the hypercube {x | 0 < x_j < L_j, j =1 to n } where {a_i}, {b_i}, c and f are constants. The boundary conditions specify the solution u along the sides x_i = 0 and x_i = L_i for i = 1 to n. Show that the eigenvalues corresponding to the eigenproblem satisfy
|\lambda| < \sum_{j=1}^n C_j cos({\pi}/{m_j}),
where {C_i} is some set of constants and {m_i} are the number of subintervals in each direction. Give the C_i explicitly in terms of the constant coefficients, along with any other conditions needed on the coefficients, and the step sizes h_i for the ith directions. Use this result to express the diagonal dominance condition in better way for m_j < \infty.
Use the Gauss-Seidel Method to find a computational approximation to the solution of
a u_{xx} + b u_{yy} + c u_x + d u_y + e u + f = 0,
on the unit square, where {a,b} = (1.33,0.67), {c,d} = (0.53 x,-0.68 y), e = -6.3 and f = -2.6. The boundary conditions are given as{
u(x,0) = 0.2 x, u(0,y) = 0.8 y, u(x,1) = (x+4)/5., u(1,y) = (4 y + 1)/5.
Use central finite differences with steps h_i = 1/3. for each i and compute one full iterate beyond the initial iterate,
u^{(0)}(x,y) = (x + 4 y)/5.,
for interior points. Tabulate both iterates at all mesh points.

Web Source: http://www.math.uic.edu/~hanson/epp571s97.html

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MCS 571 --- EPDE Practice Problems --- Hanson Numerical Methods for PDEs

Problems:

MCS 571 --- EPDE Practice Problems --- Hanson
Numerical Methods for PDEs