Description

EX_CONVDIFF2 1D Time dependent convection and diffusion equation example.

[ FEA, OUT ] = EX_CONVDIFF2( VARARGIN ) 1D time dependent convection and diffusion equation on a line with exact solution exp(-k^2*nu*t)*sin(k*(x-a*t)) and periodic boundary conditions. Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
k           scalar {2*pi}          Simulation parameter
a           scalar {1}             Convection velocity
nu          scalar {0.1}           Diffusion coefficient
hmax        scalar {1/25}          Max grid cell size
dt          scalar {0.01}          Time step size
ischeme     scalar {2}             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 = { ...
   'k',        2*pi; ...
   'a',        1; ...
   'nu',       0.1; ...
   'hmax',     1/25; ...
   'dt'        0.01; ...
   'ischeme'   2; ...
   'sfun',     'sflag1'; ...
   'iplot',    1; ...
   'tol',      1e-1; ...
   'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


 refsol = ['exp(-',num2str(opt.k^2*opt.nu),'*t)*sin(',num2str(opt.k),'*(x-',num2str(opt.a),'*t))'];


% 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 );


 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 );


% Assembly.
 [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;
 tmax = 1;
 if( opt.ischeme==2 )       % Crank-Nicolson.
   C = [ M + dt/2*A ];
 elseif( opt.ischeme==1 )   % Backward Euler.
   C = [ M + dt*A ];
 end
 v0 = zeros(size(C,1),1);
 v0(1) =  1;
 v0(n) = -1;
 C = [C v0; v0' 0];


% Solver loop.
 while 1
   t  = t  + dt;
   it = it + 1;

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

   u1 = C\[b;0];
   u1(end) = [];

   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


% 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.
 out.err  = errnm;
 out.pass = all( errnm<opt.tol );


 if ( nargout==0 )
   clear fea out
 end