# Boundary conditions with the diffusion operator step

A tutorial of the variable diffusivity diffusion equation based on Script 36 from Trefethen's Book. This uses the LU decomposition. The steps are also timed. This is an extension of the steady_diff_bc script. Please note that this script defines functions at the end, which is only supported by MATLAB 2016b or later.

## Contents

```
clear,close all
```

## Set up grid and 2D Laplacian, boundary points included:

Notice the low number of points.

N = 64;

Use the cheb function to get the Chebyshev points.

[D,x] = cheb(N); y = x;

Create the 2D grid.

[xx,yy] = meshgrid(x,y); [xxx,yyy] = meshgrid(-1:.04:1,-1:.04:1);

## Create the differentiation matrices

Make them sparse to save space.

```
D2 = sparse(D^2);
I = speye(N+1);
% Use the outer product to create the matrices.
Dy=kron(I,D);
Dx=kron(D,I);
Dyy=kron(I,D2);
Dxx=kron(D2,I);
```

## Create variable diffusion constant

r=sqrt((xx-0.5).^2+(yy+0.75).^2); kappa0=1e-2; kappa=kappa0*(1+0.5*sech((r/0.5).^2)); kappav=kappa(:); mykappax=Dx*kappa(:); mykappay=Dy*kappa(:); kappaxv=mykappax(:); kappayv=mykappay(:); kappavmat=sparse(diag(kappav)); kappaxmat=sparse(diag(kappaxv)); kappaymat=sparse(diag(kappayv));

## Create the Laplacian operator

Combine the last two parts and build the Laplacian operator.

Llap = kappavmat*(Dxx+Dyy)+kappaxmat*Dx+kappaymat*Dy;

## Create the time evolving Laplace operator

dt=0.1; L=speye(size(Llap))-dt*Llap;

## Boundary Conditions

Impose boundary conditions by replacing appropriate rows of L:

bxp1 = find(xx==1); % Rows of the big matrix at which x=1. bxn1 = find(xx==-1); % Rows of the big matrix at which x=-1. byp1 = find(yy==1); % Rows of the big matrix at which y=1. byn1 = find(yy==-1); % Rows of the big matrix at which y=-1. rhs = zeros((N+1)^2,1); L(bxp1,:) = zeros(N+1,(N+1)^2); L(bxp1,bxp1) = eye(N+1); rhs(bxp1)=-yy(bxp1); % Dirichlet BC at x=-1 notice how RHS is modified. for ii=1:length(byp1) L(byp1(ii),:) = Dy(byp1(ii),:); rhs(byp1)=2.5; % Set L=Dy at y=1, notice RHS is set to zero above. L(bxn1(ii),:) = Dx(bxn1(ii),:); rhs(bxn1)=sin(pi*(1+yy(bxn1))); % Set L=Dy at x=-1, notice RHS modifed. end L(byn1,:) = zeros(N+1,(N+1)^2); L(byn1,byn1) = eye(N+1); rhs(byn1)=-cos(3*pi*xx(byn1));

## Solve the equation

Use spparms to show the logic used by Matlab to solve the linear system with the "\" command.

```
spparms('spumoni',2)
```

Just solve the equations with nothing extra.

tic u1 = L\rhs; uu1 = reshape(u1,N+1,N+1); justsolve=toc

sp\: bandwidth = 4160+1+4095. sp\: is A diagonal? no. sp\: is band density (0.0291729) > bandden (0.5) to try banded solver? no. sp\: is A triangular? no. sp\: is A morally triangular? no. sp\: is A a candidate for Cholesky (symmetric, real positive diagonal)? no. sp\: use Unsymmetric MultiFrontal PACKagewith automatic reordering. UMFPACK V5.4.0 (May 20, 2009), Control: Matrix entry defined as: double Int (generic integer) defined as: UF_long 0: print level: 2 1: dense row parameter: 0.2 "dense" rows have > max (16, (0.2)*16*sqrt(n_col) entries) 2: dense column parameter: 0.2 "dense" columns have > max (16, (0.2)*16*sqrt(n_row) entries) 3: pivot tolerance: 0.1 4: block size for dense matrix kernels: 32 5: strategy: 0 (auto) 6: initial allocation ratio: 0.7 7: max iterative refinement steps: 2 12: 2-by-2 pivot tolerance: 0.01 13: Q fixed during numerical factorization: 0 (auto) 14: AMD dense row/col parameter: 10 "dense" rows/columns have > max (16, (10)*sqrt(n)) entries Only used if the AMD ordering is used. 15: diagonal pivot tolerance: 0.001 Only used if diagonal pivoting is attempted. 16: scaling: 1 (divide each row by sum of abs. values in each row) 17: frontal matrix allocation ratio: 0.5 18: drop tolerance: 0 19: AMD and COLAMD aggressive absorption: 1 (yes) The following options can only be changed at compile-time: 8: BLAS library used: Fortran BLAS. size of BLAS integer: 8 9: compiled for MATLAB 10: CPU timer is POSIX times ( ) routine. 11: compiled for normal operation (debugging disabled) computer/operating system: Linux size of int: 4 UF_long: 8 Int: 8 pointer: 8 double: 8 Entry: 8 (in bytes) sp\: UMFPACK's factorization was successful. sp\: UMFPACK's solve was successful. UMFPACK V5.4.0 (May 20, 2009), Info: matrix entry defined as: double Int (generic integer) defined as: UF_long BLAS library used: Fortran BLAS. size of BLAS integer: 8 MATLAB: yes. CPU timer: POSIX times ( ) routine. number of rows in matrix A: 4225 number of columns in matrix A: 4225 entries in matrix A: 520449 memory usage reported in: 16-byte Units size of int: 4 bytes size of UF_long: 8 bytes size of pointer: 8 bytes size of numerical entry: 8 bytes strategy used: symmetric ordering used: amd on A+A' modify Q during factorization: no prefer diagonal pivoting: yes pivots with zero Markowitz cost: 130 submatrix S after removing zero-cost pivots: number of "dense" rows: 0 number of "dense" columns: 0 number of empty rows: 0 number of empty columns 0 submatrix S square and diagonal preserved pattern of square submatrix S: number rows and columns 4095 symmetry of nonzero pattern: 1.000000 nz in S+S' (excl. diagonal): 508032 nz on diagonal of matrix S: 4095 fraction of nz on diagonal: 1.000000 AMD statistics, for strict diagonal pivoting: est. flops for LU factorization: 1.79750e+10 est. nz in L+U (incl. diagonal): 10523205 est. largest front (# entries): 6230016 est. max nz in any column of L: 2496 number of "dense" rows/columns in S+S': 0 symbolic factorization defragmentations: 0 symbolic memory usage (Units): 1195977 symbolic memory usage (MBytes): 18.2 Symbolic size (Units): 10846 Symbolic size (MBytes): 0 symbolic factorization CPU time (sec): 0.05 symbolic factorization wallclock time(sec): 0.05 matrix scaled: yes (divided each row by sum of abs values in each row) minimum sum (abs (rows of A)): 1.00000e+00 maximum sum (abs (rows of A)): 4.30992e+03 symbolic/numeric factorization: upper bound actual % variable-sized part of Numeric object: initial size (Units) 1089912 1085688 100% peak size (Units) 15802496 11432659 72% final size (Units) 8154726 5295040 65% Numeric final size (Units) 8177995 5316197 65% Numeric final size (MBytes) 124.8 81.1 65% peak memory usage (Units) 15872229 11502392 72% peak memory usage (MBytes) 242.2 175.5 72% numeric factorization flops 3.65049e+10 1.79750e+10 49% nz in L (incl diagonal) 7186015 5271844 73% nz in U (incl diagonal) 8386756 5263908 63% nz in L+U (incl diagonal) 15568546 10531527 68% largest front (# entries) 12892006 6230016 48% largest # rows in front 3149 2496 79% largest # columns in front 4094 2496 61% initial allocation ratio used: 0.851 # of forced updates due to frontal growth: 0 number of off-diagonal pivots: 0 nz in L (incl diagonal), if none dropped 5271844 nz in U (incl diagonal), if none dropped 5263908 number of small entries dropped 0 nonzeros on diagonal of U: 4225 min abs. value on diagonal of U: 1.00e-02 max abs. value on diagonal of U: 1.00e+00 estimate of reciprocal of condition number: 1.00e-02 indices in compressed pattern: 50236 numerical values stored in Numeric object: 10523335 numeric factorization defragmentations: 1 numeric factorization reallocations: 1 costly numeric factorization reallocations: 1 numeric factorization CPU time (sec): 9.62 numeric factorization wallclock time (sec): 2.78 numeric factorization mflops (CPU time): 1868.51 numeric factorization mflops (wallclock): 6465.84 symbolic + numeric CPU time (sec): 9.67 symbolic + numeric mflops (CPU time): 1858.84 symbolic + numeric wall clock time (sec): 2.83 symbolic + numeric mflops (wall clock): 6351.60 solve flops: 4.68820e+07 iterative refinement steps taken: 1 iterative refinement steps attempted: 1 sparse backward error omega1: 1.84e-16 sparse backward error omega2: 0.00e+00 solve CPU time (sec): 0.05 solve wall clock time (sec): 0.05 solve mflops (CPU time): 937.64 solve mflops (wall clock time): 937.64 total symbolic + numeric + solve flops: 1.80219e+10 total symbolic + numeric + solve CPU time: 9.72 total symbolic + numeric + solve mflops (CPU): 1854.11 total symbolic+numeric+solve wall clock time: 2.88 total symbolic+numeric+solve mflops(wallclock) 6257.61 justsolve = 2.8828

Solve the equations with factoring.

tic [ll,uu,pp,qq]= lu(L); factortime=toc tic u2 = qq*(uu\(ll\(pp*rhs))); uu2 = reshape(u2,N+1,N+1); factorsolve=toc

UMFPACK V5.4.0 (May 20, 2009), Control: Matrix entry defined as: double Int (generic integer) defined as: UF_long 0: print level: 2 1: dense row parameter: 0.2 "dense" rows have > max (16, (0.2)*16*sqrt(n_col) entries) 2: dense column parameter: 0.2 "dense" columns have > max (16, (0.2)*16*sqrt(n_row) entries) 3: pivot tolerance: 0.1 4: block size for dense matrix kernels: 32 5: strategy: 0 (auto) 6: initial allocation ratio: 0.7 7: max iterative refinement steps: 2 12: 2-by-2 pivot tolerance: 0.01 13: Q fixed during numerical factorization: 0 (auto) 14: AMD dense row/col parameter: 10 "dense" rows/columns have > max (16, (10)*sqrt(n)) entries Only used if the AMD ordering is used. 15: diagonal pivot tolerance: 0.001 Only used if diagonal pivoting is attempted. 16: scaling: 0 (no) 17: frontal matrix allocation ratio: 0.5 18: drop tolerance: 0 19: AMD and COLAMD aggressive absorption: 1 (yes) The following options can only be changed at compile-time: 8: BLAS library used: Fortran BLAS. size of BLAS integer: 8 9: compiled for MATLAB 10: CPU timer is POSIX times ( ) routine. 11: compiled for normal operation (debugging disabled) computer/operating system: Linux size of int: 4 UF_long: 8 Int: 8 pointer: 8 double: 8 Entry: 8 (in bytes) UMFPACK V5.4.0 (May 20, 2009), Info: matrix entry defined as: double Int (generic integer) defined as: UF_long BLAS library used: Fortran BLAS. size of BLAS integer: 8 MATLAB: yes. CPU timer: POSIX times ( ) routine. number of rows in matrix A: 4225 number of columns in matrix A: 4225 entries in matrix A: 520449 memory usage reported in: 16-byte Units size of int: 4 bytes size of UF_long: 8 bytes size of pointer: 8 bytes size of numerical entry: 8 bytes strategy used: symmetric ordering used: amd on A+A' modify Q during factorization: no prefer diagonal pivoting: yes pivots with zero Markowitz cost: 130 submatrix S after removing zero-cost pivots: number of "dense" rows: 0 number of "dense" columns: 0 number of empty rows: 0 number of empty columns 0 submatrix S square and diagonal preserved pattern of square submatrix S: number rows and columns 4095 symmetry of nonzero pattern: 1.000000 nz in S+S' (excl. diagonal): 508032 nz on diagonal of matrix S: 4095 fraction of nz on diagonal: 1.000000 AMD statistics, for strict diagonal pivoting: est. flops for LU factorization: 1.79750e+10 est. nz in L+U (incl. diagonal): 10523205 est. largest front (# entries): 6230016 est. max nz in any column of L: 2496 number of "dense" rows/columns in S+S': 0 symbolic factorization defragmentations: 0 symbolic memory usage (Units): 1195977 symbolic memory usage (MBytes): 18.2 Symbolic size (Units): 10846 Symbolic size (MBytes): 0 symbolic factorization CPU time (sec): 0.05 symbolic factorization wallclock time(sec): 0.04 matrix scaled: no symbolic/numeric factorization: upper bound actual % variable-sized part of Numeric object: initial size (Units) 1089912 1085688 100% peak size (Units) 15802496 11432659 72% final size (Units) 8154726 5295040 65% Numeric final size (Units) 8177995 5314084 65% Numeric final size (MBytes) 124.8 81.1 65% peak memory usage (Units) 15872229 11500279 72% peak memory usage (MBytes) 242.2 175.5 72% numeric factorization flops 3.65049e+10 1.79750e+10 49% nz in L (incl diagonal) 7186015 5271844 73% nz in U (incl diagonal) 8386756 5263908 63% nz in L+U (incl diagonal) 15568546 10531527 68% largest front (# entries) 12892006 6230016 48% largest # rows in front 3149 2496 79% largest # columns in front 4094 2496 61% initial allocation ratio used: 0.851 # of forced updates due to frontal growth: 0 number of off-diagonal pivots: 0 nz in L (incl diagonal), if none dropped 5271844 nz in U (incl diagonal), if none dropped 5263908 number of small entries dropped 0 nonzeros on diagonal of U: 4225 min abs. value on diagonal of U: 1.00e+00 max abs. value on diagonal of U: 1.87e+03 estimate of reciprocal of condition number: 5.34e-04 indices in compressed pattern: 50236 numerical values stored in Numeric object: 10523335 numeric factorization defragmentations: 1 numeric factorization reallocations: 1 costly numeric factorization reallocations: 1 numeric factorization CPU time (sec): 12.96 numeric factorization wallclock time (sec): 5.73 numeric factorization mflops (CPU time): 1386.96 numeric factorization mflops (wallclock): 3137.00 symbolic + numeric CPU time (sec): 13.01 symbolic + numeric mflops (CPU time): 1381.63 symbolic + numeric wall clock time (sec): 5.77 symbolic + numeric mflops (wall clock): 3115.26 factortime = 8.5793 sp\: bandwidth = 4195+1+0. sp\: is A diagonal? no. sp\: is A triangular? yes. sp\: do a triangular solve. sp\: bandwidth = 0+1+4067. sp\: is A diagonal? no. sp\: is A triangular? yes. sp\: do a triangular solve. factorsolve = 0.0386

The difference between the solutions.

figure(1) clf pcolor(xx,yy,uu1-uu2), shading interp, colorbar, title('Difference between solution without and with factoring')

The individual solutions.

figure(2) clf subplot(2,1,1) pcolor(xx,yy,uu1),shading interp,caxis([-1 1]) subplot(2,1,2) pcolor(xx,yy,uu2),shading interp,caxis([-1 1])

The diffusivity.

figure(3) clf pcolor(xx,yy,kappa),shading flat,colorbar, xlabel('x') ylabel('y')

Create the derivatives.

u1y=Dy*u1; u1x=Dx*u1; uy=reshape(u1y,N+1,N+1); ux=reshape(u1x,N+1,N+1);

figure(4) clf % Plot the derivatives. subplot(2,2,1) pcolor(xx,yy,ux),shading interp,caxis([-10 10]) subplot(2,2,2) pcolor(xx,yy,uy),shading interp,caxis([-10 10]) subplot(2,2,3) % Plot the derivatives at the boundaries. plot(yy(bxn1),ux(bxn1),'bo-',xx(byp1),uy(byp1),'rs-') subplot(2,2,4) plot(xx(byn1),u1(byn1),'bo-',yy(bxp1),u1(bxp1),'rs-')

## Chebyshev function

CHEB compute D = differentiation matrix, x = Chebyshev grid

function [D,x] = cheb(N) if N==0, D=0; x=1; return, end x = cos(pi*(0:N)/N)'; c = [2; ones(N-1,1); 2].*(-1).^(0:N)'; X = repmat(x,1,N+1); dX = X-X'; D = (c*(1./c)')./(dX+(eye(N+1))); % off-diagonal entries D = D - diag(sum(D')); % diagonal entries end