Numerical Methods for PDEs

- 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

(a) central differencing;

(b) 3 subdivisions in both*x*and*y*;

(c) the initial iterate*U_{i,j}^{(0)}=60*for iterative methods only;

(d) tabulate answers*U_{i,j}^{(k)}*;

(e) use each of the following 3 methods:

(1) explicit with FGE and BS;

(2) SOR with*\omega=1.5*for 2 iterations beyond the initial iterate;

(3) PRADI method with*\rho=2.5**(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.

{Partial ans.:

Explicit:*[U_{2,2},U_{3,2},U_{2,3},U_{3,3}] =(4ch) [45.81,56.14,79.11,82.78]*;

SOR:*[U_{2,2}^{(2)},...,U_{3,3}^{(2)}] =(4ch) [45.09,65.13,82.38,85.27]*;

PRADI:*[UY_{1,2},UY_{2,2},UY_{3,2},UY_{4,2}] =(4ch) [51.77,73.35,57.85,76.62]*in column order} - 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.

{Ans. strongly depends on how*\rho*is defined.} - Use the SOR method with
*\omega=1.3*on the above EPDE PRADI problem for two full iterates beyond the initial one.-
{Partial ans. assumes

*(1-\omega)*is coefficient of*U_{i,j}^{(k)}*term in SOR equation;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).-
{Final answer only (the initial iterate is k=0):

j U_{1,j}^{(3)} U_{2,j}^{(3)} U_{3,j}^{(3)} U_{4,j}^{(3)} U_{5,j}^{(3)} 5 .2500 .3750 .5000 .6250 .1125e1 4 .1875 .2964(3) .5625(3) .8532(1) 1.375 3 .1250 .2352(1) .4752(1,0) .8273(2) 1.250 2 .06250 .1784(3) .3760(5,4) .6801(7,9) 1.125 1 0 .1250 .2500 .3750 .7500

[Note that (number) denotes a legimate alternate digit;*||D||_{\infty}=66>||L+U||_{\infty}=64*in the*\infty*-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*{*where**x**| 0 < x_j < L_j, j =1 to n }*{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*i*th 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.

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