FEATool Multiphysics  v1.16.5 Finite Element Analysis Toolbox
ex_convdiff4.m File Reference

## 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--' );
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