Description

EX_DIFFUSION2 1D Time dependent diffusion equation examples.

[ FEA, OUT ] = EX_DIFFUSION2( VARARGIN ) 1D time dependent diffusion equation on a line with exact solutions. Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
nu          scalar {1e-1}          Diffusion coefficient
icase       scalar {2}             Test case
hmax        scalar {1/25}          Max grid cell size
dt          scalar {0.1}           Time step size
ischeme     scalar {3}             Time stepping scheme
sfun        string {sflag1}        Shape function
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { ...
   'nu',       1e-1; ...
   'icase',    2; ...
   'hmax',     1/25; ...
   'dt'        0.1; ...
   'ischeme'   3; ...
   'sfun',     'sflag1'; ...
   'iplot',    1; ...
   'tol',      1e-2; ...
   'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


 tmax = 1;
 switch( opt.icase )
   case 1
     refsol = '(1-x)*2.2+x*3.3';
   case 2
     n = 1;
     refsol = ['exp(-',num2str(opt.nu*n^2*pi^2),'*t)*sin(',num2str(n*pi),'*x)'];
 end


% Grid generation.
 fea.grid = linegrid( 1/opt.hmax, 0, 1 );


% Problem definition.
 fea.sdim  = { 'x' };
 fea = addphys( fea, @convectiondiffusion );
 fea.phys.cd.sfun = { opt.sfun };
 fea.phys.cd.eqn.coef{2,4} = { opt.nu };
 fea = parsephys(fea);


% Parse and solve problem.
 x = fea.grid.p';
 n = length(x);
 if( strcmp( opt.sfun,'sflag2' ) )
   x = [ x; (x(2:end)+x(1:end-1))/2 ];
 end
 t   = 0;
 u0  = eval( refsol );
 fea = parseprob( fea );
 if( opt.ischeme>0 )

   fea.bdr.d{1} = refsol;
   fea.bdr.d{2} = refsol;
   fea.bdr.n    = cell(1,2);
   [fea.sol.u,tlist] = solvetime( fea, 'fid', fid, 'init', u0, 'ischeme', opt.ischeme, 'tstep', opt.dt, 'tmax', tmax, 'nstbwe', 0, 'tstop', 0, 'imass', 2 );

 else

   [M,A,f] = assembleprob( fea, 'f_m', 1, 'imass', 1, 'f_a', 1, 'f_f', 1, 'f_sparse', 1 );
   M = spdiags( full(sum(M')'), 0, size(M,1), size(M,1) );
   fea.sol.u = u0;
   dt = opt.dt;
   it = 0;
   tlist = 0;
   if( opt.ischeme==-1 )       % Crank-Nicolson.
     C = [ M + dt/2*A ];
   elseif( opt.ischeme==-2 )   % Backward Euler.
     C = [ M + dt*A ];
   end
   C(1,:) = 0;
   C(1,1) = 1;
   C(n,:) = 0;
   C(n,n) = 1;
   while 1
     t  = t  + dt;
     it = it + 1;

     u_r = eval( refsol );
     if( opt.ischeme==-1 )       % Crank-Nicolson.
       b = [ [ M - dt/2*A ]*u0 + dt*f ];
     elseif( opt.ischeme==-2 )   % Backward Euler.
       b = [ M*u0 + dt*f ];
     end
     b(1) = u_r(1);
     b(n) = u_r(n);

     u1 = C\b;

     if( t>=tmax )
       break
     end

     err = norm( u1 - u_r )/norm( u_r );
     errnm(it) = err;
     if( ~isempty(fid) )
       fprintf( fid, 'Time = %f, error norm = %d\n', t, err );
     end

     fea.sol.u = [ fea.sol.u u1 ];
     tlist = [ tlist t ];
     u0 = u1;
   end

 end


% Postprocessing.
 if( opt.iplot>0 )
   figure;
   if( opt.iplot>1 )
     i_sol_list = 1:numel(tlist);
   else
     i_sol_list = numel(tlist);
   end
   [~,ix] = sort( x );
   for i_sol=i_sol_list
     t = tlist(i_sol);
     clf
     postplot( fea, 'surfexpr', 'c', 'solnum', i_sol );
     hold on
     u_r = eval( refsol );
     plot( sort(x), u_r(ix), 'r--' );
     title( ['Solution at time ',num2str(t)])
     xlabel( 'x' )
     drawnow
   end
 end


% Error checking.
 for i_sol=1:numel(tlist)
   u_i = fea.sol.u(:,i_sol);
   t   = tlist(i_sol);
   u_r = eval( refsol );
   errnm(i_sol) = norm( u_i - u_r )/norm( u_r );
 end
 out.err  = errnm;
 out.pass = all( errnm<opt.tol );


 if ( nargout==0 )
   clear fea out
 end