Description

EX_CONVDIFF4 1D Burgers equation (convection and diffusion) example.

[ FEA, OUT ] = EX_CONVDIFF4( VARARGIN ) 1D Burgers equation with steady solution, u_t + (b*u-c)*u_x - nu*u_xx = 0 with exact solution c/b*(1-tanh(c/(2*nu)*(x-x0))). Tests both time dependent and steady solvers. Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
b           scalar {1.0}           Strength of nonlinearity
c           scalar {0.13}          Convection velocity
nu          scalar {0.01}          Diffusion coefficient
x0          scalar {0.5}           Posision of smooth shock
hmax        scalar {1/25}          Max grid cell size
ischeme     scalar {-1}            Solver scheme (<0 = stationary)
nsolve      scalar {2}             Nonlinear solver (when ischeme<0)
dt          scalar {0.1}           Time step size
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 = { ...
             'b',        1.0; ...
             'c',        0.13; ...
             'nu',       0.01; ...
             'x0',       0.5; ...
             'hmax',     1/20; ...
             'ischeme'   -1; ...
             'nsolve'    2; ...
             'dt'        0.1; ...
             'sfun',     'sflag1'; ...
             'iplot',    1; ...
             'tol',      1e-2; ...
             'fid',      1 };
 [got,opt] = parseopt(cOptDef,varargin{:});
 fid       = opt.fid;


 b  = opt.b;
 c  = opt.c;
 nu = opt.nu;
 x0 = opt.x0;
 refsol = [num2str(c/b),'*(1-tanh(',num2str(c/(2*nu)),'*(x-',num2str(x0),')))'];
 bcd1   = c/b*(1-tanh((c/(2*nu)*(0-x0))));
 bcd2   = c/b*(1-tanh((c/(2*nu)*(1-x0))));

% 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} = { [num2str(b),'*c-',num2str(c)] };
 fea = parsephys(fea);


% Parse and solve problem.
 fea = parseprob( fea );
 fea.bdr.d{1} = bcd1;
 fea.bdr.d{2} = bcd2;
 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
 if( opt.ischeme<0 )
   init = 0;
   jac.form  = {[1;1]};
   jac.coef  = {[num2str(b),'*cx']};
   fea.sol.u = solvestat( fea, 'fid', fid, 'init', init, 'maxnit', 1000, 'nsolve', 2, 'jac', jac, 'nsolve', opt.nsolve );
 else
   init = [num2str(bcd1),'+x*',num2str(bcd2-bcd1)];
   [fea.sol.u,tlist] = solvetime( fea, 'fid', fid, 'init', init, 'ischeme', opt.ischeme, 'tstep', opt.dt, 'tmax', 10 );
 end


% Postprocessing.
 if( opt.iplot>0 )
   figure
   i_sol = size(fea.sol.u,2);
   [~,ix] = sort( x );
   postplot( fea, 'surfexpr', 'c', 'solnum', i_sol );
   hold on
   u_r = real( eval( refsol ) );
   plot( sort(x), u_r(ix), 'r--' );
   title( 'Steady solution' )
   xlabel( 'x' )
   drawnow
 end


% Error checking.
 i_sol = size(fea.sol.u,2);
 u_i   = fea.sol.u(:,i_sol);
 u_r   = real( eval( refsol ) );
 errnm = norm( u_i - u_r )/norm( u_r );
 out.err  = errnm;
 out.pass = all( errnm<opt.tol );


 if ( nargout==0 )
   clear fea out
 end