Description

EX_CONVDIFF3 1D Time dependent convection and diffusion equation example.

[ FEA, OUT ] = EX_CONVDIFF3( VARARGIN ) 1D time dependendt convection and diffusion equation on a line with an infinately oscillating exact solution. Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
w           scalar {1.5*pi}        Simulation parameter
U           scalar {2}             Simulation parameter
a           scalar {1}             Convection velocity
nu          scalar {0.1}           Diffusion coefficient
hmax        scalar {1/25}          Max grid cell size
dt          scalar {0.02}          Time step size
ischeme     scalar {3}             Time stepping scheme
sfun        string {sflag2}        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 = { ...
   'w',        1.5*pi; ...
   'U',        2; ...
   'a',        1; ...
   'nu',       0.1; ...
   'hmax',     1/25; ...
   'dt'        0.02; ...
   'ischeme'   3; ...
   'sfun',     'sflag1'; ...
   'iplot',    1; ...
   'tol',      3e-2; ...
   'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


 w  = opt.w;
 U  = opt.U;
 a  = opt.a;
 nu = opt.nu;
 l1 = ['(',num2str(a),'+sqrt(',num2str(a^2),'+',num2str(4*nu*w),'*i))/',num2str(2*nu)];
 l2 = ['(',num2str(a),'-sqrt(',num2str(a^2),'+',num2str(4*nu*w),'*i))/',num2str(2*nu)];
 refsol  = ['(exp(',l1,'*x)-exp(',l2,'*x))/(exp(',l1,')-exp(',l2,'))*',num2str(U),'*exp(i*',num2str(w),'*t)'];
 refsol0 = ['(exp(',l1,'*x)-exp(',l2,'*x))/(exp(',l1,')-exp(',l2,'))*',num2str(U)];


% 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.phys.cd.eqn.coef{3,4} = { opt.a  };
 fea = parsephys(fea);


% Parse and solve problem.
 fea = parseprob( fea );
 fea.bdr.d{1} = 0;
 fea.bdr.d{2} = [num2str(U),'*cos(',num2str(w),'*t)'];
 fea.bdr.n    = cell(1,2);
 x = fea.grid.p';
 if( strcmp( opt.sfun,'sflag2' ) )
   x = [ x; (x(2:end)+x(1:end-1))/2 ];
 end
 init = real( eval( refsol0 ) );
 [fea.sol.u,tlist] = solvetime( fea, 'fid', fid, 'init', init, 'ischeme', opt.ischeme, 'tstep', opt.dt, 'tmax', 1 );


% 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 = real( 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 = real( 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